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www.hg1398.com:高中数学苏教版必修5分层测评:2.2.1&2.2.2.2等差数列的性质(含答案)

学业分层测评(八) (建议用时:45 分钟) [学业达标] 一、填空题 1.在△ ABC 中,三内角 A,B,C 成等差数列,则角 B 等于 . 【解析】 ∵A,B,C 成等差数列,∴B 是 A,C 的等差中项,则有 A+C=2B,又∵A +B+C=180° , ∴3B=180° ,从而 B=60° . 【答案】 60° 2.已知 a= 1 1 ,b= ,则 a,b 的等差中项是 3+ 2 3- 2 1 = 3- 2, 3+ 2 . 【解析】 因为 a= b= a+b 1 = 3+ 2,所以 = 3. 2 3- 2 3 . 【答案】 3.在等差数列{an}中,已知 a2+a3+a10+a11=36,则 a5+a8= 【解析】 由等差数列的性质,可得 a5+a8=a3+a10=a2+a11,∴36=2(a5+a8),故 a5 +a8=18. 【答案】 18 4.设数列{an},{bn}都是等差数列,若 a1+b1=7,a3+b3=21,则 a5+b5= . 【导学号:92862040】 21-7 14 【解析】 ∵{an},{bn}都是等差数列,∴{an+bn}也是等差数列,其公差为 = 2 2 =7, ∴a5+b5=7+(5-1)× 7=35. 【答案】 35 1 5 1 5.若等差数列的前三项依次是 , , ,那么这个数列的第 101 项是 6 x+1 x x 5 1 1 【解析】 由已知得 2× = + , 6x x+1 x 1 1 解得 x=2,∴a1= ,d= , 3 12 1 1 2 ∴a101= +100× =8 . 3 12 3 . 2 【答案】 8 3 6 .已知等差数列 {an} 的公差为 d(d≠0) ,且 a3 + a6 + a10 + a13 = 32 ,若 am = 8 ,则 m = . 【解析】 由等差数列性质 a3+a6+a10+a13=(a3+a13)+(a6+a10)=2a8+2a8=4a8=32, ∴a8=8,又 d≠0,∴m=8. 【答案】 8 a2-a1 7. 已知数列-1, a1, a2, -4 与数列 1, b1, b2, b3, -5 各自成等差数列, 则 = b2 -4- - 【解析】 设数列-1,a1,a2,-4 的公差是 d,则 a2-a1=d= 4- 1 -5+1 a2-a1 1 b2= =-2,故知 = . 2 b2 2 【答案】 1 2 . . =-1, 8.已知数列{an}为等差数列且 a1+a7+a13=4π,则 tan(a2+a12)= 4π 【解析】 由等差数列的性质得 a1+a7+a13=3a7=4π, ∴a7= , ∴tan(a2+a12)=tan(2a7) 3 =tan 8π 2π =tan =- 3. 3 3 【答案】 - 3 二、解答题 b+c a+c a+b 1 1 1 9.已知 , , 成等差数列,求证: , , 也成等差数列. a b c a b c 1 1 1 2 1 1 【证明】 ∵ , , 成等差数列,∴ = + , a b c b a c 即 2ac=b(a+c). ∵ b+c a+b c b+c +a a+b + = a c ac 2 = c2+a2+b a+c ac = a2+c2+2ac = ac a+c b a+c ∴ = a+c . b b+c a+c a+b , , 成等差数列. a b c 10.若三个数 a-4,a+2,26-2a 适当排列后构成递增等差数列,求 a 的值和相应的数 列. 【解】 显然 a-4<a+2, (1)若 a-4,a+2,26-2a 成等差数列,则(a-4)+(26-2a)=2(a+2),∴a=6,相应的 等差数列为:2,8,14. (2)若 a-4,26-2a,a+2 成等差数列, 则(a-4)+(a+2)=2(26-2a), ∴a=9,相应的等差数列为:5,8,11. (3)若 26-2a,a-4,a+2 成等差数列,则(26-2a)+(a+2)=2(a-4), ∴a=12,相应的等差数列为:2,8,14. [能力提升] 1 1.在等差数列{an}中,若 a2+a4+a6+a8+a10=80,则 a7- a8 的值为 2 【解析】 ∵a2+a10=a4+a8=2a6, ∴a2+a4+a6+a8+a10=5a6=80,∴a6=16, 1 1 1 1 ∴a7- a8=(a6+d)- (a6+2d)= a6= × 16=8. 2 2 2 2 【答案】 8 2. 《九章算术》 “竹九节”问题: 现有一根 9 节的竹子, 自上而下各节的容积成等差数列, 上面 4 节的容积共 3 升,下面 3 节的容积共 4 升,则第 5 节的容积为 升. . 【导学号:92862041】 ?a1+a2+a3+a4=3, ? 【解析】 设最上面一节的容积为 a1,公差为 d,则有? ?a7+a8+a9=4, ? ?4a1+6d=3, ? 即? ? ?3a1+21d=4, ?a =22, 解得? 7 ?d=66, 1 13 67 67 则 a5= ,故第 5 节的容积为 升. 66 66 【答案】 67 66 3.在如下数表中,已知每行、每列中的数都成等差数列, 那么位于表中的第 n 行第 n+1 列的数是 . 【解析】 第 n 行的第一个数是 n,第 n 行的数构成以 n 为公差的等差数列,则其第 n +1 项为 n+n· n=n2+n. 【答案】 n2+n an 4.已知{an}满足 a1=1,且 an+1= (n∈N*). 3an+1 ?1? (1)求证:数列?a ?是等差数列; ? n? (2)求数列{an}的通项公式. 【解】 (1)证明:∵an+1= ∴ 即 3an+1 1 = =3+ , a a an+1 n n 1 1 1 - =3. an+1 an an , 3an+1 ?1? 1 即?a ?是首项为 =1,公差为 3

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