無法求出common ratio- @ rdfvjur的部落格

標題:

無法求出common ratio?

發問:

The sum of the first 5 terms and the sum of first 10 terms of a geometric sequence are 31 and 1023 respectively. Find the first term and the common ratio of the sequence.

最佳解答:

20160602 Method 2： ∵ The sum of the first 5 terms = 31 a + ar + ar2 + ar3 + ar? = 31 ...... ① ∵ The sum of first 10 terms = 1023 a + ar + ar2 + ar3 + ar? + ar? + ar? + ar? + ar? + ar? = 1023 ...... ② ② - ①： ar? + ar? + ar? + ar? + ar? = 992 (a + ar + ar2 + ar3 + ar?)r? = 992 31r? = 992 ...... ① r? = 32 r = 2 Sub r = 2 into ① a + 2a + 4a + 8a + 16a = 31 a = 1 ∴ The first term = 1 and the common ratio = 2 ???????????????????????????????????????????????????????????????????????????????????????????????? 20160602 Method 1： ∵ The sum of the first 5 terms = 31 a(r? - 1)/(r - 1) = 31 ...... ① ∵ The sum of first 10 terms = 1023 a(r1? - 1)/(r - 1) = 1023 ...... ② ② / ①： (r1? - 1) / (r? - 1) = 33 r1? - 1 = 33r? - 33 r1? - 33r? + 32 = 0 (r? - 32)(r? - 1) = 0 r? = 32orr = 1 ( rejected ) r = 2 Sub r = 2 into ① a(2? - 1)/(2 - 1) = 31 a = 1 ∴ The first term = 1 and the common ratio = 2 Remark： If r = 1, the sum of the first 5 terms ≠ 31 and the sum of first 10 terms ≠ 1023 ???????????????????????????????????????????????????????????????????????????????????????????????? Geometric Sequence： a, ar, ar2, ar3, ... , ar??1 The sum of the first n terms = a + ar + ar2 + ar3 + ... + ar??1 ...... [ Here are the first n terms but not n - 1 terms ] = a(r? - 1)/(r - 1) ...... for r ≠ 1 ???????????????????????????????????????????????????????????????????????????????????????????????? 相片：我們發生某些問題，請再試一次。