mathematical induction

let P(n) be the statement
"8^n + 2*7^n - 1 is divisible by 7" for all positive integers n
when n=1
8^1+2*7^1-1=21 which is divisible by 7
assume when n=k, P(k) is true
8^k + 2*7^k - 1 is divisible by 7
let 8^k + 2*7^k - 1 =7M where M is an integer
when n=k+1
8^(k+1) + 2*7^(k+1) - 1
=8 ( 8^k + 2*7^k - 1 )+7^k+7
=8(7M)+7(7^(k-1)+1)
=7(8M+7^(k-1)+1)
which is divisible by 7
when n=k+1, P(k+1) is true
By mathematical induction , for all positive integers n ,8^n + 2*7^n - 1 is divisible by 7

2007-02-03 01:41:32 補充：
原來我做錯了﹐你問得好呀when n=k+18^(k+1) + 2*7^(k+1) - 1=8 ( 8^k + 2*7^k - 1 )-2*7^k+7=8(7M)-7(2*7^(k-1)-1)=7(8M-2*7^(k-1)+1)which is divisible by 7我覺得你重新發問的問題那裡個2個答案都有些問題

put n=1
8^n + 2×7^n - 1
=8+2×7-1
=21
∵21 is divisible by 7
∴8^n + 2×7^n - 1 is divisible by 7

put n=k
8^k + 2×7^k - 1
let 8^k + 2×7^k - 1 =7S where S is an integer

put n=k+1
8^(k+1) + 2×7^(k+1) - 1
=8 ( 8^k + 2×7^k - 1 )+7^k+7
=8(7S)+7(7^(k-1)+1)
=7(8S+7^(k-1)+1)
which is divisible by 7

By mathematical induction , for all positive integers n ,8^n + 2*7^n - 1 is divisible by 7