Question

Wn

求解這題🙏🙏

9" (9-1) = 9² × 40
(29) 設S=1+2+22+23+...+2400,若S是n位整數,且最高位數字是k,則數對(n,k)=
0
答案:(121,5)
/0g/0+10g/100+109/0000+109/00000000

Answer 1

✨ ベストアンサー ✨

qn

S是等比級數
先用等比級數公式計算 S 值：
首項 = 1
公比 = 2
項數 = 401
S = 1 × (2⁴⁰¹-1)/(2-1)
= 2⁴⁰¹-1

取常用對數找首數、尾數
logS = log(2⁴⁰¹-1)
≈ log(2⁴⁰¹)
= 401 log(2)
≈ 401 × 0.3010
= 120.7010
S ≈ 10¹²⁰·⁷⁰¹
= 10⁰·⁷⁰¹ × 10¹²⁰
因為 log5 ≈ 1 - 0.3010 = 0.6990
log6 ≈ 0.3010 + 0.4771 = 0.7781
所以 5 < 10⁰·⁷⁰¹ < 6

S = 5.⋯ × 10¹²⁰
所以 n = 121, k = 5