S(n)=a(rn-1)/r-1

問題一 :6+(-12)+24)+(-48)+96

首項 a = 6 ,
公比 r = -12/6 = - 2
項數 n = 5

S(n) = a(r^n - 1)/(r - 1)

= 6[(-2)^5 - 1] / (- 2 - 1)

= 6(-33) / (-3)

= 66

問題二:16+4+1+1/4+1/16是否用S(n)=a(rn-1)/r-1還是用什麼公式呢?

答 : 仍然用這個公式S(n)=a(r^n - 1)/(r - 1) , 因為項數有限(五項),如果項數無限, 就用 S(n) = a / (1 - r) .

a = 16
r = 4/16 = 1/4
n = 5

S(n) = 16[ (1/4)^5 - 1] / (1/4 - 1)

= 16 (1/1024 - 1) / (-3/4)

= 16( - 1023/1024) / (-3/4)

= 16*4*(-1023 / -3) / 1024

= - 341 / 16

= - 21.3125

你可以試試把 16+4+1+1/4+1/16 倒轉來看: 1/16 + 1/4 + 1 + 4 + 16,
這時 a = 1/16 , r = 4 , n = 5,

S(n) = (1/16)(4^5 - 1) / (4 - 1) = (1/16)(1023)/3 = - 341 / 16
結果一樣。