某等比數列的公比是-5/2，第ｎ項是625，首ｎ項之和是451，求ｎ的值。?

首項 = a
公比 r = -5/2

T(n) = 625
a r^(n - 1) = 625
a r^(n - 1) × r = 625 × r
a r^n = 625 × (-5/2)
a (-5/2)^n = -3125/2 ...... [1]

S(n) = 451
a (1 - r^n) / [1 - r] = 415
a (1 - r^n) / [1 - (-5/2)] = 415
a (1 - r^n) = 451 × (7/2)
a - a (-5/2)^n = 3157/2 ...... [2]

[1] + [2] :
a = 16

代入 [1] 中：
16 × (-5/2)^n = -3125/2
(-5/2)^n = -3125/32
(-5/2)^n = (-5/2)^5
n = 5 ...... (答案)