Help Induction Question (1 Viewer)

[Harkaraj]

Hi,

I would appreciate any help on the following two questions:

1. Prove by induction that 7^n + 11^n is divisible by 9 for odd n greater than or equal to 1

2. Prove by induction that 7^n + 3^n is divisible by 10 for odd n greater than or equal to 1

Thanks

 

[Trebla]

1) Test n = 1
7 + 11
= 18
= 9 x 2 hence divisible by 9
.: True for n = 1
Assume statement is true for n = k (where k is an odd integer)
i.e. 7^k + 11^k = 9M (where M is an integer)
Need to prove the statement is true for n = k + 2 (since k is an odd integer, the next odd integer is k + 2)
i.e 7^{k + 2} + 11^{k + 2} = 9N (where N is an integer)
LHS = 7^{k + 2} + 11^{k + 2}
= 7^k.7² + 11^{k + 2}
From assumption: 7^k = 9M - 11^k
= 7²(9M - 11^k) + 11^k.11²
= 49.9M - 49.11^k + 121.11^k
= 49.9M + 72.11^k
= 9(49M + 8.11^k)
= 9N (where N = 49M + 8.11^k)
= RHS
If it is true for n = k, then it is also true for n = k + 2. Since it is true it is true for n = 1, it follows by induction that it is true for all odd integers n ≥1.

2) Test n = 1
7 + 3
= 10 x 1 hence divisible by 10
.: True for n = 1
Assume statement is true for n = k (where k is an odd integer)
i.e. 7^k + 3^k = 10P (where P is an integer)
Need to prove the statement is true for n = k + 2 (since k is an odd integer, the next odd integer is k + 2)
i.e 7^{k + 2} + 3^{k + 2} = 10Q (where Q is an integer)
LHS = 7^{k + 2} + 3^{k + 2}
= 7^k.7² + 3^{k + 2}
From assumption 7^k = 10P - 3^k
= 7².(10P - 3^k) + 3^k.3²
= 49.10P - 49.3^k + 9.3^k
= 49.10P - 40.3^k
= 10(49P - 4.3^k)
= 10Q (where Q = 49P - 4.3^k)
= RHS
If it is true for n = k, then it is also true for n = k + 2. Since it is true it is true for n = 1, it follows by induction that it is true for all odd integers n ≥1.

 

[Harkaraj]

Thanks alot for the help,

I get what I was doing wrong, i was doing (k+1) instead of (k+2) lol