無法求出common ratio?

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　20160602
Method 2：

∵ The sum of the first 5 terms = 31
a + ar + ar² + ar³ + ar⁴ = 31 ...... ①

∵ The sum of first 10 terms = 1023
a + ar + ar² + ar³ + ar⁴ + ar⁵ + ar⁶ + ar⁷ + ar⁸ + ar⁹ = 1023 ...... ②

② - ①：
ar⁵ + ar⁶ + ar⁷ + ar⁸ + ar⁹ = 992
(a + ar + ar² + ar³ + ar⁴)r⁵ = 992
31r⁵ = 992 ...... ①
r⁵ = 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(r⁵ - 1)/(r - 1) = 31 ...... ①

∵ The sum of first 10 terms = 1023
a(r¹⁰ - 1)/(r - 1) = 1023 ...... ②
② / ①：
(r¹⁰ - 1) / (r⁵ - 1) = 33
r¹⁰ - 1 = 33r⁵ - 33
r¹⁰ - 33r⁵ + 32 = 0
(r⁵ - 32)(r⁵ - 1) = 0
r⁵ = 32　　　or　　　r = 1 ( rejected )
r = 2

Sub r = 2 into ①
a(2⁵ - 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, ... , arⁿ⁻¹

The sum of the first n terms
= a + ar + ar² + ar³ + ... + arⁿ⁻¹ ...... [ Here are the first n terms but not n - 1 terms ]
= a(rⁿ - 1)/(r - 1) ...... for r ≠ 1

????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

相片：
我們發生某些問題，請再試一次。