專題08數(shù)列/r/n一、選擇題部分/r/n1/r/n./r/n(2021?高考全國甲卷/r/n?理/r/nT7)/r/n等比數(shù)列/r/n的公比為/r/nq/r/n，前/r/nn/r/n項和為/r/n，設(shè)甲：/r/n，乙：/r/n是遞增數(shù)列，則（）/r/nA.甲是乙的充分條件但不是必要條件/r/nB.甲是乙的必要條件但不是充分條件/r/nC.甲是乙的充要條件/r/nD.甲既不是乙的充分條件也不是乙的必要條件/r/nB/r/n．/r/n當(dāng)/r/n時，通過舉反例說明甲不是乙的充分條件；當(dāng)/r/n是遞增數(shù)列時，必有/r/n成立即可說明/r/n成立，則甲是乙的必要條件，即可選出答案．/r/n由題，當(dāng)數(shù)列為/r/n時，滿足/r/n，/r/n但是/r/n不是遞增數(shù)列，所以甲不是乙的充分條件．/r/n若/r/n是遞增數(shù)列，則必有/r/n成立，若/r/n不成立，則會出現(xiàn)一正一負(fù)的情況，是矛盾的，則/r/n成立，所以甲是乙的必要條件．/r/n故選/r/nB/r/n．/r/n2/r/n.(2021?浙江卷?T10)/r/n已知數(shù)列/r/n滿足/r/n./r/n記數(shù)列/r/n的前/r/nn/r/n項和為/r/n，則（）/r/nA/r/n B./r/n C./r/n D./r/nA/r/n．/r/n因為/r/n，所以/r/n，/r/n．/r/n由/r/n，即/r/n根據(jù)累加法可得，/r/n，當(dāng)且僅當(dāng)/r/n時取等號，/r/n，/r/n由累乘法可得/r/n，當(dāng)且僅當(dāng)/r/n時取等號，/r/n由裂項求和法得：/r/n所以/r/n，即/r/n．/r/n故選/r/nA/r/n．/r/n3/r/n.(2021?江蘇鹽城三模?T/r/n5/r/n)/r/n已知數(shù)列/r/neq/r/n{/r/na/r/n\/r/ns/r/n\/r/ndo/r/n(/r/nn/r/n)}/r/n的通項公式為/r/neqa/r/n\/r/ns/r/n\/r/ndo/r/n(/r/nn/r/n)/r/n＝/r/n\/r/nf/r/n(/r/nn/r/n,(/r/nn/r/n＋/r/n1)!)/r/n，則其前/r/nn/r/n項和為/r/nA/r/n．/r/neq/r/n1/r/n－/r/n\/r/nf/r/n(1,(/r/nn/r/n＋/r/n1)!)/r/nB/r/n．/r/neq/r/n1/r/n－/r/n\/r/nf/r/n(1,/r/nn/r/n!)/r/nC/r/n．/r/neq/r/n2/r/n－/r/n\/r/nf/r/n(1,/r/nn/r/n!)/r/nD/r/n．/r/neq/r/n2/r/n－/r/n\/r/nf/r/n(1,(/r/nn/r/n＋/r/n1)!)/r/nA/r/n．/r/n【考點】數(shù)列的求和：裂項相消法/r/n由題意可知，/r/neqa/r/n\/r/ns/r/n\/r/ndo/r/n(/r/nn/r/n)/r/n＝/r/n\/r/nf/r/n(/r/nn/r/n,(/r/nn/r/n＋/r/n1)!)/r/n＝/r/neq/r/n\/r/nf/r/n(/r/nn/r/n＋/r/n1/r/n－/r/n1,(/r/nn/r/n＋/r/n1)!)/r/n＝/r/neq/r/n\/r/nf/r/n(1,/r/nn/r/n!)/r/n－/r/neq/r/n\/r/nf/r/n(1,(/r/nn/r/n＋/r/n1)!)/r/n，所以/r/nS/r/nn/r/n＝/r/n1/r/n－/r/neq/r/n\/r/nf/r/n(1,2!)/r/n＋/r/neq/r/n\/r/nf/r/n(1,2!)/r/n－/r/neq/r/n\/r/nf/r/n(1,3!)/r/n＋/r/n…＋/r/neq/r/n\/r/nf/r/n(1,/r/nn/r/n!)/r/n－/r/neq/r/n\/r/nf/r/n(1,(/r/nn/r/n＋/r/n1)!)/r/n＝/r/n1/r/n－/r/neq/r/n\/r/nf/r/n(1,(/r/nn/r/n＋/r/n1)!)/r/n，故答案選/r/nA/r/n．/r/n4/r/n.(2021?江蘇鹽城三模?T/r/n10/r/n)設(shè)/r/n數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的前/r/nn/r/n項和為/r/neqS/r/n\/r/ns/r/n\/r/ndo/r/n(/r/nn/r/n)/r/n，若/r/neqa/r/n\/r/ns/r/n\/r/ndo/r/n(/r/nn/r/n)/r/n＋/r/nS/r/n\/r/ns/r/n\/r/ndo/r/n(/r/nn/r/n)/r/n＝/r/nAn/r/n\/r/ns/r/n\/r/nup/r/n6(2)/r/n＋/r/nBn/r/n＋/r/nC/r/n，則下列說法中正確的有/r/nA/r/n．存在/r/nA/r/n，/r/nB/r/n，/r/nC/r/n使得/r/n{/r/na/r/nn/r/n}/r/n是等差數(shù)列/r/nB/r/n．存在/r/nA/r/n，/r/nB/r/n，/r/nC/r/n使得/r/n{/r/na/r/nn/r/n}/r/n是等比數(shù)列/r/nC/r/n．對任意/r/nA/r/n，/r/nB/r/n，/r/nC/r/n都有/r/n{/r/na/r/nn/r/n}/r/n一定是等差數(shù)列或等比數(shù)列/r/nD/r/n．存在/r/nA/r/n，/r/nB/r/n，/r/nC/r/n使得/r/n{/r/na/r/nn/r/n}/r/n既不是等差數(shù)列也不是等比數(shù)列/r/nABD/r/n．/r/n【考點】等差與等比數(shù)列的綜合應(yīng)用/r/n由題意可知，對于選項/r/nA/r/n，取/r/nA/r/n＝/r/n0/r/n，/r/nB/r/n＝/r/nC/r/n＝/r/n1/r/n，則有/r/na/r/nn/r/n＋/r/nS/r/nn/r/n＝/r/nn/r/n＋/r/n1/r/n，此時可得到/r/na/r/nn/r/n＝/r/n1/r/n，即/r/n{/r/na/r/nn/r/n}/r/n是等差數(shù)列，所以選項/r/nA/r/n正確；對于選項/r/nB/r/n，取/r/nA/r/n＝/r/n0/r/n，/r/nB/r/n＝/r/n0/r/n，/r/nC/r/n＝/r/n1/r/n，則有/r/na/r/nn/r/n＋/r/nS/r/nn/r/n＝/r/n1/r/n，所以/r/nn/r/n≥/r/n2/r/n時，/r/na/r/nn/r/n－/r/n1/r/n＋/r/nS/r/nn/r/n－/r/n1/r/n＝/r/n1/r/n，兩式相減可得/r/n2/r/na/r/nn/r/n＝/r/na/r/nn/r/n－/r/n1/r/n，即數(shù)列/r/n{/r/na/r/nn/r/n}/r/n是等比數(shù)列，所以選項/r/nB/r/n正確；對于選項/r/nCD/r/n，取/r/nA/r/n＝/r/nC/r/n＝/r/n0/r/n，/r/nB/r/n＝/r/n2/r/n，則有/r/na/r/nn/r/n＋/r/nS/r/nn/r/n＝/r/n2/r/nn/r/n，所以/r/nn/r/n≥/r/n2/r/n時，/r/na/r/nn/r/n－/r/n1/r/n＋/r/nS/r/nn/r/n－/r/n1/r/n＝/r/n2(/r/nn/r/n－/r/n1)/r/n，兩式相減可得/r/na/r/nn/r/n＝/r/nEQ/r/n\/r/nF/r/n(1,2)/r/na/r/nn/r/n－/r/n1/r/n＋/r/n1/r/n，即/r/na/r/nn/r/n－/r/n2/r/n＝/r/nEQ/r/n\/r/nF/r/n(1,2)/r/n(/r/na/r/nn/r/n－/r/n1/r/n－/r/n2)/r/n，即數(shù)列/r/n{/r/na/r/nn/r/n－/r/n2}/r/n是以/r/nEQ/r/n\/r/nF/r/n(1,2)/r/n為公比的等比數(shù)列，所以/r/n{/r/na/r/nn/r/n}/r/n既不是等差數(shù)列也不是等比數(shù)列，所以選項/r/nC/r/n錯誤，選項/r/nD/r/n正確；綜上，答案選/r/nABD/r/n．/r/n5/r/n.(2021?河南鄭州三模?理T/r/n5/r/n)/r/n已知等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的公差不為零，且/r/na/r/n3/r/n2/r/n＝/r/na/r/n1/r/na/r/n7/r/n，/r/nS/r/nn/r/n為其前/r/nn/r/n項和，則/r/n＝（）/r/nA/r/n．/r/n /r/nB/r/n．/r/n /r/nC/r/n．/r/n /r/nD/r/n．/r/nn/r/n（/r/nn/r/n﹣/r/n1/r/n）/r/nA/r/n．/r/n設(shè)等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的公差為/r/nd/r/n≠/r/n0/r/n，∵/r/na/r/n3/r/n2/r/n＝/r/na/r/n1/r/na/r/n7/r/n，/r/n∴/r/n＝/r/na/r/n1/r/n（/r/na/r/n1/r/n+6/r/nd/r/n），化為：/r/na/r/n1/r/n＝/r/n2/r/nd/r/n，/r/n∴/r/nS/r/nn/r/n＝/r/nna/r/n1/r/n+/r/n×/r/n＝/r/na/r/n1/r/n，/r/n則/r/n＝/r/n．/r/n6/r/n.(2021?河南焦作三模?理T/r/n4/r/n)/r/n已知公比大于/r/n1/r/n的等比數(shù)列/r/n{/r/na/r/nn/r/n}/r/n滿足/r/na/r/n2/r/na/r/nm/r/n＝/r/na/r/n6/r/na/r/nn/r/n，/r/na/r/nm/r/n2/r/n＝/r/na/r/n6/r/na/r/n10/r/n，則/r/nm/r/n+/r/nn/r/n＝（）/r/nA/r/n．/r/n4/r/n /r/nB/r/n．/r/n8/r/n /r/nC/r/n．/r/n12/r/n /r/nD/r/n．/r/n16/r/nC/r/n．/r/n∵/r/na/r/n2/r/na/r/nm/r/n＝/r/na/r/n6/r/na/r/nn/r/n，/r/na/r/nm/r/n2/r/n＝/r/na/r/n6/r/na/r/n10/r/n，公比/r/nq/r/n＞/r/n1/r/n，/r/n∴由等比數(shù)列的性質(zhì)可得：/r/nm/r/n＝/r/n8/r/n，/r/nn/r/n＝/r/n4/r/n，/r/n∴/r/nm/r/n+/r/nn/r/n＝/r/n12/r/n．/r/n7/r/n.(2021?重慶名校聯(lián)盟三模?T/r/n6/r/n．)在/r/n我國古代著名的數(shù)學(xué)專著《九章算術(shù)》里有一段敘述：今有良馬與駑馬發(fā)長安至齊，齊去長安一千一百二十五里，良馬初日行一百零三里，日增一十三里；駑馬初日行九十七里，日減半里；良馬先至齊，復(fù)還迎駑馬，二馬相逢．問：幾日相逢？（）/r/nA/r/n．/r/n8/r/n日/r/n /r/nB/r/n．/r/n9/r/n日/r/n /r/nC/r/n．/r/n12/r/n日/r/n /r/nD/r/n．/r/n16/r/n日/r/nB/r/n．/r/n由題可知，良馬每日行程/r/na/r/nn/r/n構(gòu)成一個首項為/r/n103/r/n，公差/r/n13/r/n的等差數(shù)列，/r/n駑馬每日行程/r/nb/r/nn/r/n構(gòu)成一個首項為/r/n97/r/n，公差為﹣/r/n0.5/r/n的等差數(shù)列，/r/n則/r/na/r/nn/r/n＝/r/n103+13/r/n（/r/nn/r/n﹣/r/n1/r/n）＝/r/n13/r/nn/r/n+90/r/n，/r/nb/r/nn/r/n＝/r/n97/r/n﹣/r/n0.5/r/n（/r/nn/r/n﹣/r/n1/r/n）＝/r/n97.5/r/n﹣/r/n0.5/r/nn/r/n，/r/n則數(shù)列/r/n{/r/na/r/nn/r/n}/r/n與數(shù)列/r/n{/r/nb/r/nn/r/n}/r/n的前/r/nn/r/n項和為/r/n1125/r/n×/r/n2/r/n＝/r/n2250/r/n，/r/n又∵數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的前/r/nn/r/n項和為/r/n×（/r/n103+13/r/nn/r/n+90/r/n）＝/r/n×（/r/n193+13/r/nn/r/n），/r/n數(shù)列/r/n{/r/nb/r/nn/r/n}/r/n的前/r/nn/r/n項和為/r/n×（/r/n97+97.5/r/n﹣/r/n0.5/r/nn/r/n）＝/r/n×（/r/n194.5/r/n﹣/r/nn/r/n），/r/n∴/r/n×（/r/n193+13/r/nn/r/n）/r/n+/r/n×（/r/n194.5/r/n﹣/r/nn/r/n）＝/r/n2250/r/n，/r/n整理得：/r/n25/r/nn/r/n2/r/n+775/r/nn/r/n﹣/r/n9000/r/n＝/r/n0/r/n，即/r/nn/r/n2/r/n+31/r/nn/r/n﹣/r/n360/r/n＝/r/n0/r/n，/r/n解得：/r/nn/r/n＝/r/n9/r/n或/r/nn/r/n＝﹣/r/n40/r/n（舍），即九日相逢．/r/n8/r/n.(2021?安徽蚌埠三模?文T/r/n4/r/n．)/r/n已知等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的前/r/nn/r/n項和為/r/nS/r/nn/r/n，/r/nS/r/n1/r/n＝/r/n1/r/n，/r/nS/r/n5/r/n＝/r/n25/r/n，則/r/n＝（）/r/nA/r/n．/r/n3/r/n /r/nB/r/n．/r/n6/r/n /r/nC/r/n．/r/n9/r/n /r/nD/r/n．/r/n12/r/nA/r/n．/r/n因為等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n中，/r/na/r/n1/r/n＝/r/nS/r/n1/r/n＝/r/n1/r/n，/r/n所以/r/nS/r/n5/r/n＝/r/n5+10/r/nd/r/n＝/r/n25/r/n，/r/n所以/r/nd/r/n＝/r/n2/r/n，則/r/n＝/r/na/r/n1/r/n+/r/nd/r/n＝/r/n3/r/n．/r/n9/r/n.(2021?貴州畢節(jié)三模?文T/r/n9/r/n．)如/r/n圖，有甲、乙、丙三個盤子和放在甲盤子中的四塊大小不相同的餅，按下列規(guī)則把餅從甲盤全部移到乙盤中：/r/n①/r/n每次只能移動一塊餅；/r/n②/r/n較大的餅不能放在較小的餅上面，則最少需要移動的次數(shù)為（）/r/nA/r/n．/r/n7/r/n /r/nB/r/n．/r/n8/r/n /r/nC/r/n．/r/n15/r/n /r/nD/r/n．/r/n16/r/nC/r/n．/r/n假設(shè)甲盤中有/r/nn/r/n塊餅，從甲盤移動到乙盤至少需要/r/na/r/nn/r/n次，則/r/na/r/n1/r/n＝/r/n1/r/n，/r/n當(dāng)/r/nn/r/n≥/r/n2/r/n時，可先將較大的餅不動，將剩余的/r/nn/r/n﹣/r/n1/r/n塊餅先移動到丙盤中，至少需要移動/r/na/r/nn/r/n﹣/r/n1/r/n次，再將最大的餅移動到乙盤，需要移動/r/n1/r/n次，/r/n最后將丙盤中所有的丙移動到乙盤中，至少需要移動/r/na/r/nn/r/n﹣/r/n1/r/n次，/r/n由上可知，/r/na/r/nn/r/n＝/r/n2/r/na/r/nn/r/n﹣/r/n1/r/n+1/r/n，且/r/na/r/n1/r/n＝/r/n1/r/n，/r/n所以/r/na/r/n2/r/n＝/r/n2/r/na/r/n1/r/n+1/r/n＝/r/n3/r/n，/r/na/r/n3/r/n＝/r/n2/r/na/r/n2/r/n+1/r/n＝/r/n7/r/n，/r/na/r/n4/r/n＝/r/n2/r/na/r/n3/r/n+1/r/n＝/r/n15/r/n，/r/n則最少需要移動的次數(shù)為/r/n15/r/n次．/r/n1/r/n0/r/n.(2021?貴州畢節(jié)三模?文T/r/n5/r/n．)/r/n“干支紀(jì)年法”是中國歷法上自古以來使用的紀(jì)年方法，甲、乙、丙、丁、戊、己、庚、辛、壬、癸被稱為“十天干”，子、丑、寅、卯、辰、巳、午、未、申、酉、戌、亥叫做“十二地支”．“天干”以“甲”字開始，“地支”以“子”字開始，兩者按干支順序相配，組成了干支紀(jì)年法，其相配順序為：甲子、乙丑、丙寅、…、癸酉，甲戌、乙亥、丙子、…、癸未，甲申、乙酉、丙戌、…、癸巳，…，/r/n共得到/r/n60/r/n個組合，稱六十甲子，周而復(fù)始，無窮無盡．/r/n2021/r/n年是“干支紀(jì)年法”中的辛丑年，那么/r/n2015/r/n年是“干支紀(jì)年法”中的（）/r/nA/r/n．甲辰年/r/n /r/nB/r/n．乙巳年/r/n /r/nC/r/n．丙午年/r/n /r/nD/r/n．乙未年/r/nD/r/n．/r/n由題意可知，甲、乙、丙、丁、戊、己、庚、辛、壬、癸被稱為“十天干”，/r/n子、丑、寅、卯、辰、巳、午、未、申、酉、戌、亥叫做“十二地支”，/r/n2021/r/n年是“干支紀(jì)年法”中的辛丑年，則/r/n2020/r/n年為庚子，/r/n2019/r/n年為己亥，/r/n2018/r/n年為戊戌，/r/n2017/r/n年為丁酉，/r/n2016/r/n年為丙申，/r/n2015/r/n年為乙未．/r/n1/r/n1/r/n.(2021?遼寧朝陽三模?T/r/n4/r/n．)跑/r/n步是一項有氧運動，通過跑步，我們能提高肌力，同時提高體內(nèi)的基礎(chǔ)代謝水平，加速脂肪的燃燒，養(yǎng)成易瘦體質(zhì)．小林最近給自己制定了一個/r/n200/r/n千米的跑步健身計劃，他第一天跑了/r/n8/r/n千米，以后每天比前一天多跑/r/n0.5/r/n千米，則他要完成該計劃至少需要（）/r/nA/r/n．/r/n16/r/n天/r/n /r/nB/r/n．/r/n17/r/n天/r/n /r/nC/r/n．/r/n18/r/n天/r/n /r/nD/r/n．/r/n19/r/n天/r/nB/r/n．/r/n設(shè)需要/r/nn/r/n天完成計劃，由題意易知每天跑步的里程為，以/r/n8/r/n為首項，/r/n0.5/r/n為公差的等差數(shù)列，∴/r/n，/r/n∴/r/nn/r/n2/r/n+31/r/nn/r/n﹣/r/n800/r/n≥/r/n0/r/n，當(dāng)/r/nn/r/n＝/r/n16/r/n時，/r/n16/r/n2/r/n+31/r/n×/r/n16/r/n﹣/r/n800/r/n＜/r/n0/r/n，/r/n當(dāng)/r/nn/r/n＝/r/n17/r/n時，/r/n17/r/n2/r/n+17/r/n×/r/n31/r/n﹣/r/n800/r/n＞/r/n0/r/n．/r/n1/r/n2/r/n.(2021?河南濟(jì)源平頂山許昌三模?文T/r/n4/r/n．)“/r/n干支紀(jì)年法”是我國歷法的一種傳統(tǒng)紀(jì)年法，甲、乙、丙、丁、戊、己、庚、辛、壬、癸被稱為”十天干”；子、丑、寅、卯、辰、巳、午、未、申、西、戌、亥叫做“十二地支”．“天干”以“甲”字開始，“地支”以“子”字開始，兩者按干支順序相配，組成了干支紀(jì)年法，其相配順序為甲子、乙丑、丙寅、……癸酉；甲戌、乙亥、丙子、…、癸未；甲申、乙酉、丙戌、…、癸巳；…，共得到/r/n60/r/n個組合，稱六十甲子，周而復(fù)始，無窮無盡．/r/n2021/r/n年是“干支紀(jì)年法”中的辛丑年，那么/r/n2121/r/n年是“干支紀(jì)年法”中的（）/r/nA/r/n．庚午年/r/n /r/nB/r/n．辛未年/r/n /r/nC/r/n．庚辰年/r/n /r/nD/r/n．辛巳年/r/nD/r/n．/r/n天干：甲、乙、丙、丁、戊、己、庚、辛、壬、癸；/r/n地支：子、丑、寅、卯、辰、巳、午、未、申、酉、戌、亥，/r/n天干是以/r/n10/r/n為公差的等差數(shù)列，地支是以/r/n12/r/n為公差的等差數(shù)列，/r/n2021/r/n年是“干支紀(jì)年法”中的辛丑年，則/r/n2121/r/n的天干為辛，地支為巳/r/n．/r/n1/r/n3/r/n.(2021?安徽宿州三模?理T/r/n8/r/n．)/r/n各項均為正數(shù)的等比數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的前/r/nn/r/n項和為/r/nS/r/nn/r/n，且/r/na/r/n1/r/na/r/n7/r/n＝/r/n3/r/na/r/n4/r/n，/r/na/r/n2/r/n與/r/na/r/n3/r/n的等差中項為/r/n18/r/n，則/r/nS/r/n5/r/n＝（）/r/nA/r/n．/r/n108/r/n /r/nB/r/n．/r/n117/r/n /r/nC/r/n．/r/n120/r/n /r/nD/r/n．/r/n121/r/nD/r/n．/r/n設(shè)等比數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的公比為/r/nq/r/n，/r/nq/r/n＞/r/n0/r/n，由/r/na/r/n1/r/na/r/n7/r/n＝/r/na/r/n4/r/n2/r/n＝/r/n3/r/na/r/n4/r/n，可得/r/na/r/n4/r/n＝/r/n3/r/n，即有/r/na/r/n1/r/nq/r/n3/r/n＝/r/n3/r/n，由/r/na/r/n2/r/n與/r/na/r/n3/r/n的等差中項為/r/n18/r/n，可得/r/na/r/n2/r/n+/r/na/r/n3/r/n＝/r/n36/r/n，即為/r/na/r/n1/r/nq/r/n+/r/na/r/n1/r/nq/r/n2/r/n＝/r/n36/r/n，解得/r/na/r/n1/r/n＝/r/n81/r/n，/r/nq/r/n＝/r/n，/r/n則/r/nS/r/n5/r/n＝/r/n＝/r/n121/r/n．/r/n1/r/n4/r/n.(2021?安徽宿州三模?文T/r/n5/r/n．)已/r/n知/r/n{/r/na/r/nn/r/n}/r/n為等差數(shù)列且/r/na/r/n1/r/n＝/r/n1/r/n，/r/na/r/n4/r/n+/r/na/r/n9/r/n＝/r/n24/r/n，/r/nS/r/nn/r/n為其前/r/nn/r/n項的和，則/r/nS/r/n12/r/n＝（）/r/nA/r/n．/r/n142/r/n /r/nB/r/n．/r/n143/r/n /r/nC/r/n．/r/n144/r/n /r/nD/r/n．/r/n145/r/nC/r/n．/r/n解法一、等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n中，設(shè)公差為/r/nd/r/n，/r/n由/r/na/r/n1/r/n＝/r/n1/r/n，/r/na/r/n4/r/n+/r/na/r/n9/r/n＝/r/n24/r/n，/r/n得（/r/na/r/n1/r/n+3/r/nd/r/n）/r/n+/r/n（/r/na/r/n1/r/n+8/r/nd/r/n）＝/r/n2/r/na/r/n1/r/n+11/r/nd/r/n＝/r/n2+11/r/nd/r/n＝/r/n24/r/n，/r/n解得/r/nd/r/n＝/r/n2/r/n，/r/n所以/r/nS/r/n12/r/n＝/r/n12/r/na/r/n1/r/n+/r/n×/r/n12/r/n×/r/n11/r/n×/r/n2/r/n＝/r/n12/r/n×/r/n1+132/r/n＝/r/n144/r/n．/r/n解法二、等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n中，/r/na/r/n1/r/n＝/r/n1/r/n，/r/na/r/n4/r/n+/r/na/r/n9/r/n＝/r/n24/r/n，/r/n所以前/r/nn/r/n項的和/r/nS/r/n12/r/n＝/r/n＝/r/n6/r/n（/r/na/r/n4/r/n+/r/na/r/n9/r/n）＝/r/n6/r/n×/r/n24/r/n＝/r/n144/r/n．/r/n1/r/n5/r/n.(2021?河南開封三模?文T/r/n7/r/n．)/r/n設(shè)數(shù)列/r/n{/r/na/r/nn/r/n}/r/n滿足/r/na/r/n1/r/n＝/r/n1/r/n，/r/n，若/r/n，則/r/nn/r/n＝（）/r/nA/r/n．/r/n4/r/n /r/nB/r/n．/r/n5/r/n /r/nC/r/n．/r/n6/r/n /r/nD/r/n．/r/n7/r/nC/r/n．/r/n根據(jù)題意，數(shù)列/r/n{/r/na/r/nn/r/n}/r/n滿足/r/na/r/n1/r/n＝/r/n1/r/n，/r/n，/r/n則數(shù)列/r/n{/r/na/r/nn/r/n}/r/n是首項/r/na/r/n1/r/n＝/r/n1/r/n，公比為/r/n的等比數(shù)列，/r/n若/r/n，即/r/na/r/n1/r/n（/r/na/r/n1/r/nq/r/n）（/r/na/r/n1/r/nq/r/n2/r/n）……（/r/na/r/n1/r/nq/r/nn/r/n﹣/r/n1/r/n）＝（/r/na/r/n1/r/n）/r/nn/r/n×/r/n＝/r/n＝/r/n，解可得：/r/nn/r/n＝/r/n6/r/n或﹣/r/n5/r/n（舍）/r/n．/r/n1/r/n6/r/n.(2021?四川瀘州三模?理T/r/n6/r/n．)已/r/n知/r/nS/r/nn/r/n為等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的前/r/nn/r/n項和，若/r/na/r/n2/r/n＝/r/n15/r/n，/r/nS/r/n5/r/n＝/r/n65/r/n，則/r/na/r/n1/r/n+/r/na/r/n4/r/n＝（）/r/nA/r/n．/r/n24/r/n /r/nB/r/n．/r/n26/r/n /r/nC/r/n．/r/n28/r/n /r/nD/r/n．/r/n30/r/nC/r/n．/r/n由題意/r/nS/r/n5/r/n＝/r/n5/r/na/r/n3/r/n＝/r/n65/r/n，/r/na/r/n3/r/n＝/r/n13/r/n，所以/r/na/r/n1/r/n+/r/na/r/n4/r/n＝/r/na/r/n2/r/n+/r/na/r/n3/r/n＝/r/n28/r/n．/r/n1/r/n7/r/n.(2021?江蘇常數(shù)三模?T/r/n12/r/n．)/r/n斐波那契，公元/r/n13/r/n世紀(jì)意大利數(shù)學(xué)家．他在自己的著作《算盤書》中記載著這樣一個數(shù)列：/r/n1/r/n，/r/n1/r/n，/r/n2/r/n，/r/n3/r/n，/r/n5/r/n，/r/n8/r/n，/r/n13/r/n，/r/n21/r/n，/r/n34/r/n，/r/n?/r/n，其中從第三個數(shù)起，每一個數(shù)都等于它前面兩個數(shù)的和，這就是著名的斐波那契數(shù)列．斐波那契數(shù)列與代數(shù)和幾何都有著不可分割的聯(lián)系．現(xiàn)有一段長為/r/na/r/n米的鐵絲，需要截成/r/nn/r/n（/r/nn/r/n＞/r/n2/r/n）段，每段的長度不小于/r/n1/r/nm/r/n，且其中任意三段都不能構(gòu)成三角形，若/r/nn/r/n的最大值為/r/n10/r/n，則/r/na/r/n的值可能是（）/r/nA/r/n．/r/n100/r/n /r/nB/r/n．/r/n143/r/n /r/nC/r/n．/r/n200/r/n /r/nD/r/n．/r/n256/r/nBC/r/n．/r/n由題意，一段長為/r/na/r/n米的鐵絲，截成/r/nn/r/n段，且其中任意三段都不能構(gòu)成三角形，/r/n當(dāng)/r/nn/r/n取最大值時，每段長度從小到大排列正好為斐波那契數(shù)列，/r/n而數(shù)列的前/r/n10/r/n項和為：/r/n1+1+2+3+5+8+13+21+34+55/r/n＝/r/n143/r/n，/r/n前/r/n11/r/n項和為：/r/n1+1+2+3+5+8+13+21+34+55+89/r/n＝/r/n232/r/n，/r/n∴只需/r/n143/r/n≤/r/na/r/n＜/r/n232/r/n，/r/nBC/r/n均符合要求．/r/n1/r/n8/r/n.(2021?上海浦東新區(qū)三模?T/r/n16/r/n．)已/r/n知函數(shù)/r/nf/r/n（/r/nx/r/n）＝/r/nsin/r/nx/r/n，各項均不相等的數(shù)列/r/n{/r/na/r/nn/r/n}/r/n滿足/r/n|/r/na/r/ni/r/n|/r/n≤/r/n（/r/ni/r/n＝/r/n1/r/n，/r/n2/r/n，…/r/nn/r/n），記/r/nG/r/n（/r/nn/r/n）＝/r/n．/r/n①/r/n若/r/na/r/nn/r/n＝（﹣/r/n）/r/nn/r/n，則/r/nG/r/n（/r/n2000/r/n）＞/r/n0/r/n；/r/n②/r/n若/r/n{/r/na/r/nn/r/n}/r/n是等差數(shù)列，且/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/n…/r/n+/r/na/r/nn/r/n≠/r/n0/r/n，則/r/nG/r/n（/r/nn/r/n）＞/r/n0/r/n對/r/nn/r/n∈/r/nN/r/n*/r/n恒成立．關(guān)于上述兩個命題，以下說法正確的是（）/r/nA/r/n．/r/n①②/r/n均正確/r/n /r/nB/r/n．/r/n①②/r/n均錯誤/r/n /r/nC/r/n．/r/n①/r/n對，/r/n②/r/n錯/r/n /r/nD/r/n．/r/n①/r/n錯，/r/n②/r/n對/r/nA/r/n．/r/nf/r/n（/r/nx/r/n）＝/r/nsin/r/nx/r/n在/r/n[/r/n﹣/r/n]/r/n上為奇函數(shù)且單調(diào)遞增，/r/n①/r/n：/r/na/r/n2/r/nk/r/n﹣/r/n1/r/n+/r/na/r/n2/r/nk/r/n＜/r/n0/r/n（/r/nk/r/n∈/r/nN/r/n*/r/n）可得/r/na/r/n2/r/nk/r/n﹣/r/n1/r/n＜﹣/r/na/r/n2/r/nk/r/n，則/r/nf/r/n（/r/na/r/n2/r/nk/r/n﹣/r/n1/r/n）＜/r/nf/r/n（﹣/r/na/r/n2/r/nk/r/n）＜/r/nf/r/n（﹣/r/na/r/n2/r/nk/r/n）＝﹣/r/nf/r/n（/r/na/r/n2/r/nk/r/n），/r/n所以/r/nf/r/n（/r/na/r/n2/r/nk/r/n﹣/r/n1/r/n）/r/n+/r/nf/r/n（/r/na/r/n2/r/nk/r/n）＜/r/n0/r/n＜/r/n0/r/n，則/r/na/r/n1/r/n+/r/na2............/r/n+/r/na/r/n2000/r/n＜/r/n0/r/n，/r/nf/r/n（/r/na/r/n1/r/n）/r/n+/r/nf/r/n（/r/na/r/n2/r/n）/r/n...../r/n.....+/r/nf/r/n（/r/na/r/n2000/r/n）/r/n+.....+/r/nf/r/n（/r/na/r/n2000/r/n）＜/r/n0/r/n，故/r/nG/r/n（/r/n2000/r/n）＞/r/n0/r/n，/r/n①/r/n正確，/r/n②/r/n：/r/n{/r/na/r/nn/r/n}/r/n為等差數(shù)列，當(dāng)/r/na/r/n1/r/n+/r/na/r/n2/r/n+.....+/r/na/r/nn/r/n＞/r/n0/r/n時，/r/n若/r/nn/r/n為偶數(shù)，/r/na/r/n＞/r/n0/r/n，/r/na/r/n1/r/n＞﹣/r/na/r/nn/r/n可得/r/nf/r/n（/r/na/r/n1/r/n）＞/r/nf/r/n（﹣/r/na/r/nn/r/n）＝﹣/r/nf/r/n（/r/na/r/nn/r/n），則/r/nf/r/n（/r/na/r/n1/r/n）/r/n+/r/nf/r/n（/r/na/r/nn/r/n）＞/r/n0/r/n，/r/n同理可得：/r/nf/r/n（/r/na/r/n2/r/n）/r/n+/r/nf/r/n（/r/na/r/nn/r/n﹣/r/n1/r/n）＞/r/n0/r/n，/r/n......./r/nf/r/n（/r/na/r/n）/r/n+/r/nf/r/n（/r/na/r/n）＞/r/n0/r/n，所以/r/nG/r/n（/r/nn/r/n）＞/r/n0/r/n，/r/n若/r/nn/r/n為奇數(shù)，/r/na/r/n1/r/n+/r/na/r/nn/r/n＝/r/na/r/n2/r/n+/r/na/r/nn/r/n﹣/r/n1/r/n＝/r/n....../r/n＝/r/n2/r/na/r/n＞/r/n0/r/n，/r/nf/r/n（/r/na/r/n1/r/n）/r/n+/r/nf/r/n（/r/na/r/nn/r/n）＞/r/n0/r/n，/r/nf/r/n（/r/na/r/n2/r/n）/r/n+/r/nf/r/n（/r/na/r/nn/r/n﹣/r/n1/r/n）＞/r/n0/r/n，/r/n...../r/n，/r/nf/r/n（/r/na/r/n）＞/r/n0/r/n，所以/r/nG/r/n（/r/nn/r/n）＞/r/n0/r/n，/r/n當(dāng)/r/na/r/n1/r/n+/r/na/r/n2/r/n+.....+/r/na/r/nn/r/n＜/r/n0/r/n時，同理可證/r/nG/r/n（/r/nn/r/n）＞/r/n0/r/n，/r/n②/r/n正確/r/n．/r/n1/r/n9/r/n.(2021?湖南三模?T/r/n5/r/n．)/r/n《周髀算經(jīng)》是我國古代的天文學(xué)和數(shù)學(xué)著作，其中有一個問題大意如下：一年有二十四個節(jié)氣，每個節(jié)氣晷長損益相同（即太陽照射物體的影子長度增加和減少的大小相同）．二十四個節(jié)氣及晷長變化如圖所示，若冬至晷長一丈三尺五寸，夏至晷長一尺五寸（注：一丈等于十尺，一尺等于十寸），則立秋晷長為（）/r/nA/r/n．五寸/r/n /r/nB/r/n．二尺五寸/r/n /r/nC/r/n．三尺五寸/r/n /r/nD/r/n．四尺五寸/r/nD/r/n．/r/n設(shè)從夏至到冬至，每個節(jié)氣晷長為/r/na/r/nn/r/n，則/r/na/r/n1/r/n＝/r/n15/r/n，冬至晷長/r/na/r/n13/r/n＝/r/n135/r/n，/r/n由題意得/r/n{/r/na/r/nn/r/n}/r/n為等差數(shù)列，則/r/nd/r/n＝/r/n＝/r/n10/r/n，故/r/na/r/n4/r/n＝/r/na/r/n1/r/n+3/r/nd/r/n＝/r/n15+30/r/n＝/r/n45/r/n．/r/n20/r/n.(2021?江西南昌三模?理T/r/n5/r/n．)已/r/n知公差不為/r/n0/r/n的等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n滿足/r/na/r/n5/r/n2/r/n+/r/na/r/n6/r/n2/r/n＝/r/na/r/n7/r/n2/r/n+/r/na/r/n8/r/n2/r/n，則（）/r/nA/r/n．/r/na/r/n6/r/n＝/r/n0/r/n /r/nB/r/n．/r/na/r/n7/r/n＝/r/n0/r/n /r/nC/r/n．/r/nS/r/n12/r/n＝/r/n0/r/n /r/nD/r/n．/r/nS/r/n13/r/n＝/r/n0/r/nC/r/n．/r/n因為公差不為/r/n0/r/n的等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n滿足/r/na/r/n5/r/n2/r/n+/r/na/r/n6/r/n2/r/n＝/r/na/r/n7/r/n2/r/n+/r/na/r/n8/r/n2/r/n，/r/n所以/r/na/r/n8/r/n2/r/n﹣/r/na/r/n5/r/n2/r/n+/r/na/r/n7/r/n2/r/n﹣/r/na/r/n6/r/n2/r/n＝/r/n0/r/n，/r/n所以（/r/na/r/n8/r/n﹣/r/na/r/n5/r/n）（/r/na/r/n8/r/n+/r/na/r/n5/r/n）/r/n+/r/n（/r/na/r/n7/r/n﹣/r/na/r/n6/r/n）（/r/na/r/n7/r/n+/r/na/r/n6/r/n）＝/r/n0/r/n，/r/n即/r/n3/r/nd/r/n（/r/na/r/n8/r/n+/r/na/r/n5/r/n）/r/n+/r/nd/r/n（/r/na/r/n7/r/n+/r/na/r/n6/r/n）＝/r/n0/r/n，/r/n因為/r/nd/r/n≠/r/n0/r/n，/r/n所以/r/n3/r/n（/r/na/r/n8/r/n+/r/na/r/n5/r/n）/r/n+/r/n（/r/na/r/n7/r/n+/r/na/r/n6/r/n）＝/r/n0/r/n，/r/n由等差數(shù)列的性質(zhì)得/r/n4/r/n（/r/na/r/n1/r/n+/r/na/r/n12/r/n）＝/r/n0/r/n，即/r/na/r/n1/r/n+/r/na/r/n12/r/n＝/r/n0/r/n，/r/n所以/r/nS/r/n12/r/n＝/r/n0/r/n．/r/n2/r/n1.(2021?江西上饒三模?理T/r/n12/r/n．)數(shù)/r/n列/r/n{/r/na/r/nn/r/n}/r/n是以/r/na/r/n為首項，/r/nq/r/n為公比的等比數(shù)列，數(shù)列/r/n{/r/nb/r/nn/r/n}/r/n滿足/r/nb/r/nn/r/n＝/r/n1+/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/n…/r/n+/r/na/r/nn/r/n（/r/nn/r/n＝/r/n1/r/n，/r/n2/r/n，…），數(shù)列/r/n{c/r/nn/r/n}/r/n滿足/r/nc/r/nn/r/n＝/r/n2+/r/nb/r/n1/r/n+/r/nb/r/n2/r/n+/r/n…/r/n+/r/nb/r/nn/r/n（/r/nn/r/n＝/r/n1/r/n，/r/n2/r/n，…）．若/r/n{c/r/nn/r/n}/r/n為等比數(shù)列，則/r/na/r/n+/r/nq/r/n＝（）/r/nA/r/n．/r/n /r/nB/r/n．/r/n3/r/n /r/nC/r/n．/r/n /r/nD/r/n．/r/n6/r/nB/r/n．/r/n數(shù)列/r/n{/r/na/r/nn/r/n}/r/n是以/r/na/r/n為首項，/r/nq/r/n為公比的等比數(shù)列，/r/na/r/nn/r/n＝/r/naq/r/nn/r/n﹣/r/n1/r/n，/r/n則/r/nb/r/nn/r/n＝/r/n1+/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/n…/r/n+/r/na/r/nn/r/n＝/r/n1+/r/n＝/r/n1+/r/n﹣/r/n，/r/n則/r/nc/r/nn/r/n＝/r/n2+/r/nb/r/n1/r/n+/r/nb/r/n2/r/n+/r/n…/r/n+/r/nb/r/nn/r/n＝/r/n2+/r/n（/r/n1+/r/n）/r/nn/r/n﹣/r/n×/r/n＝/r/n2/r/n﹣/r/n+/r/nn/r/n+/r/n，要使/r/n{c/r/nn/r/n}/r/n為等比數(shù)列，則/r/n，解得：/r/n，∴/r/na/r/n+/r/nq/r/n＝/r/n3/r/n．/r/n22/r/n.(2021?安徽馬鞍山三模?理T/r/n10/r/n．)國/r/n際數(shù)學(xué)教育大會（/r/nICME/r/n）是由國際數(shù)學(xué)教育委員會主辦的國際數(shù)學(xué)界最重要的會議，每四年舉辦一次，至今共舉辦了十三屆，第十四屆國際數(shù)學(xué)教育大會于/r/n2021/r/n年上海舉行，華東師大向全世界發(fā)出了數(shù)學(xué)教育理論發(fā)展與實踐經(jīng)驗分享的邀約，如圖甲是第七屆國際數(shù)學(xué)家大會（簡稱/r/nICME/r/n﹣/r/n7/r/n）的會徽圖案，會徽的主題圖案是由圖乙的一連串直角三角形演化而成的．/r/n其中已知：/r/nOA/r/n1/r/n＝/r/nA/r/n1/r/nA/r/n2/r/n＝/r/nA/r/n2/r/nA/r/n3/r/n＝/r/nA/r/n3/r/nA/r/n4/r/n＝/r/nA/r/n4/r/nA/r/n5/r/n＝/r/nA/r/n5/r/nA/r/n6/r/n＝/r/nA/r/n6/r/nA/r/n7/r/n＝/r/nA/r/n7/r/nA/r/n8/r/n＝/r/n?/r/n＝/r/n1/r/n，/r/nA/r/n1/r/n，/r/nA/r/n2/r/n，/r/nA/r/n3/r/n，/r/n?/r/n，為直角頂點，設(shè)這些直角三角形的周長和面積依次從小到大組成的數(shù)列分別為/r/n{/r/nl/r/nn/r/n}/r/n，/r/n{/r/nS/r/nn/r/n}/r/n，則關(guān)于此兩個數(shù)列敘述錯誤的是（）/r/nA/r/n．/r/n{/r/nS/r/nn/r/n2/r/n}/r/n是等差數(shù)列/r/n /r/nB/r/n．/r/n /r/nC/r/n．/r/n /r/nD/r/n．/r/nl/r/nn/r/n﹣/r/n1/r/n＝/r/n2/r/nS/r/nn/r/n+2/r/nS/r/nn/r/n+1/r/nC/r/n．/r/n由/r/nOA/r/n1/r/n＝/r/nA/r/n1/r/nA/r/n2/r/n＝/r/nA/r/n2/r/nA/r/n3/r/n＝/r/nA/r/n3/r/nA/r/n4/r/n＝/r/nA/r/n4/r/nA/r/n5/r/n＝/r/nA/r/n5/r/nA/r/n6/r/n＝/r/nA/r/n6/r/nA/r/n7/r/n＝/r/nA/r/n7/r/nA/r/n8/r/n＝/r/n?/r/n＝/r/n1/r/n，/r/n得/r/nOA/r/n2/r/n＝/r/n，/r/n，/r/n?/r/n，/r/n故/r/n，∴/r/nl/r/nn/r/n＝/r/nOA/r/nn/r/n+/r/nA/r/nn/r/nA/r/nn/r/n+1/r/n+/r/nOA/r/nn/r/n+1/r/n＝/r/n，/r/n①/r/nS/r/nn/r/n＝/r/n＝/r/n，/r/n對于/r/nA/r/n，/r/nS/r/nn/r/n2/r/n＝/r/n，∴/r/n{/r/nS/r/nn/r/n2/r/n}/r/n是等差數(shù)列，所以/r/nA/r/n正確；/r/n對于/r/nB/r/n，由/r/n①/r/n可知，/r/nB/r/n正確；/r/n對于/r/nC/r/n，/r/nl/r/nn/r/n﹣/r/nl/r/nn/r/n﹣/r/n1/r/n＝/r/n﹣（/r/n）＝/r/n，所以/r/nC/r/n錯誤；/r/n對于/r/nD/r/n，/r/nl/r/nn/r/n﹣/r/n1/r/n＝/r/n，/r/n2/r/nS/r/nn/r/n+2/r/nS/r/nn/r/n+1/r/n＝/r/n＝/r/nl/r/nn/r/n﹣/r/n1/r/n，所以/r/nD/r/n正確．/r/n23/r/n.(2021?安徽馬鞍山三模?文T/r/n8/r/n．)在/r/n天然氣和煤氣還未普及時，農(nóng)民通常會用水稻秸稈作為生火做飯的材料．每年水稻收割結(jié)束之后，農(nóng)民們都會把水稻秸稈收集起來，然后堆成如圖的草堆，供生火做飯使用．通常他們堆草堆的時候都是先把秸稈先捆成一捆一捆的，然后堆成下面近似成一個圓柱體，上面近似成一個圓錐體的形狀．假設(shè)圓柱體堆了/r/n7/r/n層，每層所用的小捆草數(shù)量相同，上面收小時，每層小捆草數(shù)量是下一層的/r/n倍．若共用/r/n255/r/n捆，最上一層只有一捆，則草堆自上往下共有幾層（）/r/nA/r/n．/r/n13/r/n /r/nB/r/n．/r/n12/r/n /r/nC/r/n．/r/n11/r/n /r/nD/r/n．/r/n10/r/nB/r/n．/r/n設(shè)圓錐體有/r/nn/r/n層，由題意可知最上面一層只有一捆，/r/n所以第/r/nn/r/n層有/r/n1/r/n×/r/n2/r/nn/r/n﹣/r/n1/r/n捆，圓錐體的總捆數(shù)為/r/n＝/r/n2/r/nn/r/n﹣/r/n1/r/n，/r/n圓柱體堆了/r/n7/r/n層，總捆數(shù)為/r/n7/r/n×/r/n2/r/nn/r/n﹣/r/n1/r/n，/r/n草堆的總捆數(shù)為/r/n7/r/n×/r/n2/r/nn/r/n﹣/r/n1/r/n+2/r/nn/r/n﹣/r/n1/r/n﹣/r/n2/r/nn/r/n﹣/r/n1/r/n＝/r/n255/r/n，/r/n解得/r/nn/r/n＝/r/n6/r/n，所以自下往上共有/r/n6+7/r/n﹣/r/n1/r/n＝/r/n12/r/n層/r/n．/r/n24/r/n.(2021?安徽馬鞍山三模?理T/r/n4/r/n．)已/r/n知等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n中，/r/na/r/n2/r/n+/r/na/r/n14/r/n＝/r/n18/r/n，/r/na/r/n2/r/n＝/r/n3/r/n，則/r/na/r/n10/r/n＝（）/r/nA/r/n．/r/n10/r/n /r/nB/r/n．/r/n11/r/n /r/nC/r/n．/r/n12/r/n /r/nD/r/n．/r/n13/r/nB/r/n．/r/n在等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n中，由/r/na/r/n2/r/n+/r/na/r/n14/r/n＝/r/n18/r/n，得/r/n2/r/na/r/n8/r/n＝/r/na/r/n2/r/n+/r/na/r/n14/r/n＝/r/n18/r/n，則/r/na/r/n8/r/n＝/r/n9/r/n，/r/n又/r/na/r/n2/r/n＝/r/n3/r/n，∴/r/n，∴/r/na/r/n10/r/n＝/r/na/r/n8/r/n+2/r/nd/r/n＝/r/n9+2/r/n×/r/n1/r/n＝/r/n11/r/n．/r/n25/r/n.(2021?江西九江二模?理T/r/n9/r/n．)古/r/n希臘畢達(dá)哥拉斯學(xué)派認(rèn)為數(shù)是萬物的本源，因此極為重視數(shù)的理論研究，他們常把數(shù)描繪成沙灘上的沙?；蛐∈?，并將它們排列成各種形狀進(jìn)行研究．形數(shù)就是指平面上各種規(guī)則點陣所對應(yīng)的點數(shù)，是畢哥拉斯學(xué)派最早研究的重要內(nèi)容之一．如圖是三角形數(shù)和四邊形數(shù)的前四個數(shù)，若三角形數(shù)組成數(shù)列/r/n{/r/na/r/nn/r/n}/r/n，四邊形數(shù)組成數(shù)列/r/n{/r/nb/r/nn/r/n}/r/n，記/r/nc/r/nn/r/n＝/r/n，則數(shù)列/r/n{c/r/nn/r/n}/r/n的前/r/n10/r/n項和為（）/r/nA/r/n．/r/n /r/nB/r/n．/r/n /r/nC/r/n．/r/n /r/nD/r/n．/r/nD/r/n．/r/n由題意可得，/r/n，/r/n，/r/n所以/r/n，/r/n設(shè)數(shù)列/r/n{c/r/nn/r/n}/r/n的前/r/nn/r/n項和為/r/nS/r/nn/r/n，所以/r/n，所以/r/n．/r/n26/r/n.(2021?江西九江二模?理T/r/n3/r/n．)/r/n已知等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的前/r/nn/r/n項和為/r/nS/r/nn/r/n，且滿足/r/na/r/n3/r/n＝/r/n7/r/n，/r/nS/r/n10/r/n＝/r/n20/r/n，則/r/na/r/n8/r/n＝（）/r/nA/r/n．﹣/r/n5/r/n /r/nB/r/n．﹣/r/n3/r/n /r/nC/r/n．/r/n3/r/n /r/nD/r/n．/r/n5/r/nB/r/n．/r/n設(shè)等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的公差為/r/nd/r/n，/r/n由題意得/r/n，/r/n解得/r/na/r/n1/r/n＝/r/n11/r/n，/r/nd/r/n＝﹣/r/n2/r/n，/r/n故/r/na/r/n8/r/n＝/r/n11+7/r/n×（﹣/r/n2/r/n）＝﹣/r/n3/r/n．/r/n27/r/n.(2021?浙江杭州二模?理T/r/n8/r/n．)/r/n已知數(shù)列/r/n{/r/na/r/nn/r/n}/r/n滿足/r/na/r/nn/r/n﹣/r/n1/r/n＝/r/na/r/nn/r/n+/r/na/r/nn/r/n﹣/r/n2/r/n（/r/nn/r/n≥/r/n3/r/n），設(shè)數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的前/r/nn/r/n項和為/r/nS/r/nn/r/n，若/r/nS/r/n2020/r/n＝/r/n2019/r/n，/r/nS/r/n2019/r/n＝/r/n2020/r/n，則/r/nS/r/n2021/r/n＝（）/r/nA/r/n．/r/n1008/r/n /r/nB/r/n．/r/n1009/r/n /r/nC/r/n．/r/n2016/r/n /r/nD/r/n．/r/n2018/r/nB/r/n．/r/n因為/r/na/r/nn/r/n﹣/r/n1/r/n＝/r/na/r/nn/r/n+/r/na/r/nn/r/n﹣/r/n2/r/n，（/r/nn/r/n≥/r/n3/r/n），/r/n所以/r/na/r/nn/r/n＝/r/na/r/nn/r/n+1/r/n+/r/na/r/nn/r/n﹣/r/n1/r/n，則/r/na/r/nn/r/n+1/r/n+/r/na/r/nn/r/n﹣/r/n2/r/n＝/r/n0/r/n，所以/r/na/r/nn/r/n+/r/na/r/nn/r/n+3/r/n＝/r/n0/r/n，/r/na/r/nn/r/n+3/r/n+/r/na/r/nn/r/n+6/r/n＝/r/n0/r/n，/r/n則/r/na/r/nn/r/n＝/r/na/r/nn/r/n+6/r/n，可知/r/na/r/n1/r/n+/r/na/r/n4/r/n＝/r/n0/r/n，/r/na/r/n2/r/n+/r/na/r/n5/r/n＝/r/n0/r/n，/r/na/r/n3/r/n+/r/na/r/n6/r/n＝/r/n0/r/n，/r/n所以/r/nS/r/n6/r/n＝/r/na/r/n1/r/n+/r/na/r/n2/r/n+/r/n…/r/n+/r/na/r/n6/r/n＝/r/n0/r/n，/r/n因為/r/n2019/r/n＝/r/n6/r/n×/r/n336+3/r/n，所以/r/nS/r/n2019/r/n＝/r/n0+/r/na/r/n2017/r/n+/r/na/r/n2018/r/n+/r/na/r/n2019/r/n＝/r/n2020/r/n，/r/n所以/r/na/r/n2017/r/n+/r/na/r/n2018/r/n+/r/na/r/n2019/r/n＝/r/n2020/r/n，/r/n因為/r/na/r/n2020/r/n＝/r/nS/r/n2020/r/n﹣/r/nS/r/n2019/r/n＝/r/n2019/r/n﹣/r/n2020/r/n＝﹣/r/n1/r/n，則/r/na/r/n2017/r/n＝/r/n1/r/n，/r/n所以/r/na/r/n2018/r/n+/r/na/r/n2019/r/n＝/r/n2020/r/n﹣/r/n1/r/n＝/r/n2019/r/n，因為/r/na/r/n2018/r/n＝/r/na/r/n2017/r/n+/r/na/r/n2019/r/n＝/r/n1+/r/na/r/n2019/r/n，/r/n所以/r/na/r/n2019/r/n＝/r/n1009/r/n，/r/na/r/n2018/r/n＝/r/n1010/r/n，/r/n因為/r/na/r/n2021/r/n＝/r/na/r/n2018/r/n＝﹣/r/n1010/r/n，所以/r/nS/r/n2021/r/n＝/r/nS/r/n2020/r/n+/r/na/r/n2021/r/n＝/r/n2019/r/n﹣/r/n1010/r/n＝/r/n1009/r/n．/r/n28/r/n.(2021?江西上饒二模?理T/r/n3/r/n．)等/r/n比數(shù)列/r/n{/r/na/r/nn/r/n}/r/n中，/r/na/r/n3/r/n＝/r/n4/r/n，/r/na/r/n2/r/na/r/n6/r/n＝/r/n64/r/n，則/r/na/r/n5/r/n＝（）/r/nA/r/n．/r/n /r/nB/r/n．/r/n8/r/n /r/nC/r/n．/r/n16/r/n /r/nD/r/n．/r/n32/r/nC/r/n．/r/n∵等比數(shù)列/r/n{/r/na/r/nn/r/n}/r/n中，/r/na/r/n3/r/n＝/r/n4/r/n，/r/na/r/n2/r/na/r/n6/r/n＝/r/n64/r/n，/r/n∴/r/na/r/n2/r/na/r/n6/r/n＝/r/n＝/r/n64/r/n，/r/n解得/r/na/r/n4/r/n＝±/r/n8/r/n，/r/n∴/r/nq/r/n＝/r/n＝±/r/n2/r/n，/r/n∴/r/na/r/n5/r/n＝/r/n＝/r/n4/r/n×/r/n4/r/n＝/r/n16/r/n．/r/n29/r/n.(2021?河北秦皇島二模?理T/r/n3/r/n．)南/r/n宋數(shù)學(xué)家楊輝《詳解九章算法》和《算法通變本末》中，提出垛積公式，所討論的高階等差數(shù)列前后兩項之差不相等，但是逐項差數(shù)之差或者高次差成等差數(shù)列．對這類高階等差數(shù)列的研究，在楊輝之后一般稱為“垛積術(shù)”．現(xiàn)有高階等差數(shù)列，其前/r/n6/r/n項分別/r/n1/r/n，/r/n6/r/n，/r/n13/r/n，/r/n24/r/n，/r/n41/r/n，/r/n66/r/n，則該數(shù)列的第/r/n7/r/n項為（）/r/nA/r/n．/r/n91/r/n /r/nB/r/n．/r/n99/r/n /r/nC/r/n．/r/n101/r/n /r/nD/r/n．/r/n113/r/nC/r/n．/r/n由題意得/r/n1/r/n，/r/n6/r/n，/r/n13/r/n，/r/n24/r/n，/r/n41/r/n，/r/n66/r/n的差組成數(shù)列：/r/n5/r/n，/r/n7/r/n，/r/n11/r/n，/r/n17/r/n，/r/n25/r/n…，這些數(shù)的差組成數(shù)列：/r/n2/r/n，/r/n4/r/n，/r/n6/r/n，/r/n8/r/n，/r/n10/r/n…，/r/n故該數(shù)列的第/r/n7/r/n項為/r/n10+25+66/r/n＝/r/n101/r/n．/r/n30/r/n.(2021?江西鷹潭二模?理T/r/n3/r/n．)記/r/nS/r/nn/r/n為等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的前/r/nn/r/n項和．已知/r/nS/r/n4/r/n＝/r/n0/r/n，/r/na/r/n5/r/n＝/r/n5/r/n，則（）/r/nA/r/n．/r/na/r/nn/r/n＝/r/n2/r/nn/r/n﹣/r/n5/r/n /r/nB/r/n．/r/na/r/nn/r/n＝/r/n3/r/nn/r/n﹣/r/n10/r/n /r/nC/r/n．/r/nS/r/nn/r/n＝/r/n2/r/nn/r/n2/r/n﹣/r/n8/r/nn/r/n /r/nD/r/n．/r/nS/r/nn/r/n＝/r/nn/r/n2/r/n﹣/r/n2/r/nn/r/nA/r/n．/r/n設(shè)等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的公差為/r/nd/r/n，由/r/nS/r/n4/r/n＝/r/n0/r/n，/r/na/r/n5/r/n＝/r/n5/r/n，得/r/n，∴/r/n，∴/r/na/r/nn/r/n＝/r/n2/r/nn/r/n﹣/r/n5/r/n，/r/n．/r/n3/r/n1.(2021?北京門頭溝二模?理T5)/r/n中/r/n國古代數(shù)學(xué)著作《算法統(tǒng)宗》中有這樣一個問題：/r/n“/r/n三百七十八里關(guān)，初步健步不為難，次日腳痛減一半，六朝才得到其關(guān)，要見次日行里數(shù)，請公仔細(xì)算相還/r/n./r/n”/r/n其大意為：/r/n“/r/n有一個人走/r/n378/r/n里路，第一天健步行走，從第二天起腳痛每天走的路程為前一天的一半，走了/r/n6/r/n天后到達(dá)目的地/r/n./r/n”/r/n則該人第五天走的路程為/r/n(?)/r/nA.48/r/n里/r/n /r/nB.24/r/n里/r/n /r/nC.12/r/n里/r/n /r/nD.6/r/n里/r/nC/r/n．/r/n本題考查等比數(shù)列的通項公式的運用，是基礎(chǔ)題，解題時要認(rèn)真審題，注意等比數(shù)列的性質(zhì)的合理運用．/r/n

由題意可知，每天走的路程里數(shù)構(gòu)成以/r/n1/r/n2/r/n為公比的等比數(shù)列，由/r/nS/r/n解：記每天走的路程里數(shù)為/r/n{/r/na/r/nn/r/n}/r/n，由題意知/r/n{/r/na/r/nn/r/n}/r/n是公比/r/n1/r/n2/r/n的等比數(shù)列，/r/n

由/r/nS/r/n6/r/n=378/r/n，得/r/nS/r/n6/r/n=/r/na/r/n1/r/n(1-/r/n1/r/n2/r/n6/r/n)/r/n1-/r/n1/r/n2/r/n=378/r/n，解得：/r/na/r/n1/r/n=192/r/n，/r/n∴/r/na/r/n5/r/n=192×/r/n1/r/n2/r/n4/r/n=12(/r/n里/r/n)./r/n故選：/r/nC./r/n

/r/n32/r/nA/r/n．當(dāng)/r/nk/r/n＝/r/n時，數(shù)列/r/n{/r/na/r/nn/r/n}/r/n為遞減數(shù)列/r/n /r/nB/r/n．當(dāng)/r/nk/r/n＝/r/n時，數(shù)列/r/n{/r/na/r/nn/r/n}/r/n一定有最大項/r/n /r/nC/r/n．當(dāng)/r/n0/r/n＜/r/nk/r/n＜/r/n時，數(shù)列/r/n{/r/na/r/nn/r/n}/r/n為遞減數(shù)列/r/n /r/nD/r/n．當(dāng)/r/n為正整數(shù)時，數(shù)列/r/n{/r/na/r/nn/r/n}/r/n必有兩項相等的最大項/r/nBCD/r/n．/r/na/r/nn/r/n＜/r/na/r/nn/r/n+1/r/n?/r/nn/r/n?/r/nk/r/nn/r/n＜（/r/nn/r/n+1/r/n）?/r/nk/r/nn/r/n+1/r/n?/r/nn/r/n＜（/r/nn/r/n+1/r/n）/r/nk/r/n?/r/n，/r/na/r/nn/r/n＞/r/na/r/nn/r/n+1/r/n?/r/nn/r/n?/r/nk/r/nn/r/n＞（/r/nn/r/n+1/r/n）?/r/nk/r/nn/r/n+1/r/n?/r/nn/r/n＞（/r/nn/r/n+1/r/n）/r/nk/r/n?/r/n，/r/n對于/r/nA/r/n，因為/r/nk/r/n＝/r/n，所以/r/na/r/n1/r/n＝/r/n，/r/na/r/n2/r/n＝/r/n2/r/n＝/r/n，于是/r/na/r/n1/r/n＝/r/na/r/n2/r/n，所以/r/nA/r/n錯；/r/n對于/r/nB/r/n，因為/r/nk/r/n＝/r/n，所以/r/n＝/r/n4/r/n，于是當(dāng)/r/nn/r/n＞/r/n4/r/n時，/r/n{/r/na/r/nn/r/n}/r/n遞減，所以數(shù)列/r/n{/r/na/r/nn/r/n}/r/n一定有最大項，所以/r/nB/r/n對；/r/n對于/r/nC/r/n，因為當(dāng)/r/n0/r/n＜/r/nk/r/n＜/r/n時，/r/n＜/r/n，所以當(dāng)/r/nn/r/n≥/r/n1/r/n＞/r/n＞/r/n時，數(shù)列/r/n{/r/na/r/nn/r/n}/r/n為遞減數(shù)列，所以/r/nC/r/n對；/r/n對于/r/nD/r/n，設(shè)/r/n＝/r/nm/r/n，當(dāng)/r/nn/r/n＞/r/nm/r/n，即/r/nn/r/n≥/r/nm/r/n+1/r/n時數(shù)列/r/n{/r/na/r/nn/r/n}/r/n為遞減，當(dāng)/r/nn/r/n＜/r/nm/r/n時/r/n{/r/na/r/nn/r/n}/r/n為遞增，/r/n，最大項為/r/na/r/nm/r/n＝/r/n，/r/na/r/nm/r/n+1/r/n＝（/r/nm/r/n+1/r/n）/r/n＝/r/n，/r/n所以數(shù)列/r/n{/r/na/r/nn/r/n}/r/n必有兩項相等的最大項，所以/r/nD/r/n對．/r/n33/r/n.(2021?安徽淮北二模?文T/r/n8/r/n．)若/r/n正項等比數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的公比為/r/ne/r/n（/r/ne/r/n是自然對數(shù)的底數(shù)），則數(shù)列/r/n{/r/nlna/r/n2/r/nn/r/n﹣/r/n1/r/n}/r/n是（）/r/nA/r/n．公比為/r/ne/r/n2/r/n的等比數(shù)列/r/n /r/nB/r/n．公比為/r/n2/r/n的等比數(shù)列/r/n /r/nC/r/n．公差為/r/n2/r/ne/r/n的等差數(shù)列/r/n /r/nD/r/n．公差為/r/n2/r/n的等差數(shù)列/r/nD/r/n．/r/n正項等比數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的公比為/r/ne/r/n（/r/ne/r/n是自然對數(shù)的底數(shù)），/r/n∴/r/na/r/n2/r/nn/r/n﹣/r/n1/r/n＝/r/n，∴/r/nlna/r/n2/r/nn/r/n﹣/r/n1/r/n＝/r/n＝/r/nlna/r/n1/r/n+2/r/nn/r/n﹣/r/n2/r/n＝/r/n2/r/nn/r/n+/r/n（/r/nlna/r/n1/r/n﹣/r/n2/r/n），/r/n∴數(shù)列/r/n{/r/nlna/r/n2/r/nn/r/n﹣/r/n1/r/n}/r/n是公差為/r/n2/r/n的等差數(shù)列．/r/n34/r/n.(2021?吉林長春一模?文T/r/n11./r/n)如/r/n圖/r/n,/r/n在面積為/r/n1/r/n的正方形/r/n內(nèi)做四邊形/r/n使/r/n以此類推,在四邊形/r/n內(nèi)再做四邊形/r/n……，/r/n記四邊形/r/n的面積為/r/n，/r/n則/r/nB/r/n．/r/n由圖可知/r/n所以其前/r/n項和為/r/n,/r/n故選/r/nB./r/n35/r/n.(2021?寧夏銀川二模?文T/r/n6/r/n．)記/r/nS/r/nn/r/n為等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n的前/r/nn/r/n項和，已知/r/na/r/n2/r/n＝/r/n0/r/n，/r/na/r/n6/r/n＝/r/n8/r/n，則/r/nS/r/n10/r/n＝（）/r/nA/r/n．/r/n66/r/n /r/nB/r/n．/r/n68/r/n /r/nC/r/n．/r/n70/r/n /r/nD/r/n．/r/n80/r/nC/r/n．/r/n等差數(shù)列/r/n{/r/na/r/nn/r/n}/r/n中，/r/na/r/n2/r/n＝/r/n0/r/n，/r/na/r/n6/r/n＝/r/n8/r/n，故/r/nd/r/n＝/r/n＝/r/n2/r/n，/r/na/r/n1/r/n＝﹣/r/n2/r/n，則/r/nS/r/n10/r/n＝/r/n10/r/n×（﹣/r/n2/r/n）/r/n+45/r/n×/r/n2/r/n＝/r/n70/r/n．/r/n二、填空題部分/r/n36/r/n.(2021?新高考全國Ⅰ卷?T1/r/n6/r/n)/r/n某校學(xué)生在研究民間剪紙藝術(shù)時，發(fā)現(xiàn)剪紙時經(jīng)