An=5^n+2*3^n-1+1(n属于正整数)能被8整除数学归纳法证明

Question

An=5^n+2*3^n-1+1(n属于正整数)能被8整除 数学归纳法证明

Answer 1

n=1
a1=5+2+1=8 is divisible by 8
Assume p(k) is true
ie
ak = 5^k+ 2(3^(k-1))+1 = 8m ( where m is integer )
when n=k+1
a(k+1)
= 5^(k+1)+ 2(3^(k))+1
= 5(5^k+2(3^(k-1)+1) + 2(3^(k))-10(3^(k-1))-4
=40m+ 2(3^k) - (6+4)3^(k-1) -4
=40m - 4(3^(k-1))-4
=4(20m-3^(k-1)-1)
= 4(2m') ( 3^(k-1) is odd number )
=8m'
is divisible by 8
By principle of MI,it is true for all n