2.1.2 数列的通项公式与递推公式（学案）-【成才之路】2020-2021学年高中新课程数学同步学习指导（人教A版必修5）

数学 （必修 5·人教 A 版）　 此方程无正整数解,故 419 不是数列中的项. 令 n(n + 1) = 420, ∴ n2 + n - 420 = 0, ∴ (n - 20)(n + 21) = 0, ∵ n∈N + ,∴ n = 20. 故 420 是数列中的第 20 项. 第 2 课时　 数列的通项公式与递推公式 新知导学 　 　 1. an - 1 　 递推公式 预习自测 1. (1) × 　 递推公式也是给出数列的一种重要方法. (2) × 　 并不是所有的数列都有递推公式. 例如 2 精确到 1,0. 1, 0. 01,0. 001,…的不足近似值排列成一列数:1,1. 4,1. 41,1. 414,…就 没有递推公式. (3) × 　 通过图象,列表和通项公式我们可以确定一个数列,另外根据 递推公式,并且知道数列的第一项(或前 n 项),也可以确定数列. (4)√　 因为 a1 = - 1,an + 1 = an - 2,所以 a2 = - 1 - 2 = - 3,a3 = - 3 - 2 = - 5. 2. 5　 ∵ xn + 1 = f(xn ), ∴ x1 = f(x0 ) = f(5) = 2, x2 = f(x1 ) = f(2) = 1, x3 = f(x2 ) = f(1) = 5, x4 = f(x3 ) = f(5) = 2, x2 018 = x2 015 = … = x2 = 1, ∴ x2 019 = f(x2 018 ) = f(1) = 5. 3. 2 19 　 a2 = a1 1 + 3a1 = 2 1 + 6 = 2 7 , a3 = a2 1 + 3a2 = 2 7 1 + 6 7 = 2 13 , a4 = a3 1 + 3a3 = 2 13 1 + 6 13 = 2 19 . 4. 31　 a3 = 6 + a1 = 7,a5 = 6 + a3 = 13, a7 = 6 + a5 = 19,a9 = 6 + a7 = 25, a11 = 6 + a9 = 31. 5. 8 5 　 a2 = 1 + 1 = 2,a3 = 1 + 1 a2 = 3 2 , a4 = 1 + 1 a3 = 5 3 ,a5 = 1 + 1 a4 = 8 5 . 互动探究解疑 　 　 典例试做 1:(1)∵ a1 = 1,an + 1 = n n + 1 an , ∴ a2 = 1 1 + 1 × 1 = 1 2 ,a3 = 2 1 + 2 × 1 2 = 1 3 , a4 = 3 1 + 3 × 1 3 = 1 4 ,a5 = 4 1 + 4 × 1 4 = 1 5 . (2)猜想:an = 1 n . 　 　 跟踪练习 1:(1)D　 a22 - a1 a3 = ( - 1) 1 , ∵ a1 = 1,a2 = 3. ∴ 9 - a3 = - 1,解得 a3 = 10. a23 - a2 a4 = ( - 1) 2 ,100 - 3a4 = 1,∴ 3a4 = 99,∴ a4 = 33. (2) 因 为 an = an - 1 + 1 n + 1 + n ( n ≥ 2 ), 所 以 an - an - 1 = 1 n + 1 + n = n + 1 - n, 所以 a1 = 1, a2 - a1 = 3 - 2, a3 - a2 = 4 - 3, a4 - a3 = 5 - 4, …… an - an - 1 = n + 1 - n. 所以 an = a1 + (a2 - a1 ) + (a3 - a2 ) + (a4 - a3 ) + … + (an - an - 1 ) = 1 + ( 3 - 2) + ( 4 - 3) + ( 5 - 4) + ( n + 1 - n) = n + 1 - 2 + 1, 当 n = 1 时,a1 符合上式,所以 an = n + 1 - 2 + 1. 　 　 典例试做 2:(1) 80 　 因为 anan + 1 an + 2 an + 3 = an + an + 1 + an + 2 + an + 3 ,所以 an + 1 an + 2 an + 3 an + 4 = an + 1 + an + 2 + an + 3 + an + 4 . 两式相减得,(an - an + 4 )(an + 1 an + 2 an + 3 - 1) = 0, 由于 an + 1 an + 2 an + 3 ≠1,所以 an = an + 4 , 又 a1 = a2 = 1,a3 = 2,所以 a4 = 4, a1 + a2 + a3 + a4 = 8, 所以 a1 + a2 + a3 + … + a40 = 10(a1 + a2 + a3 + a4 ) = 80.