Plzzz prove by mathematical induction?

Question

Using Mathematical Induction prove that 10^(2n-1) + 1 is divisible by 11.

Answer 1

10^(2n-1) + 1 is divisible by 11
if n = 1
LHS = 10^(2n-1) + 1 = 10^(1) + 1 = 11 which is divisible.

Assuming it is true for n = k that is
10^(2k-1) + 1 is divisible by 11
Now for n = k+1 the expression is
10^(2k+1) + 1 = 100.10^(2k-1) + 1
= 100.10^(2k-1) + 100 - 99
= 100.(10^(2k-1)+1) - 99
Now 10^(2k-1)+1 is divisible by 11 and 99 is divisible by 11 so 100.(10^(2k-1)+1) - 99 is divisble by 11
so 10^(2k+1) + 1 is divisble by 11.

Hence proved by induction

Answer 2

1. Show it's true for n=1. I think you can do that.
2. Show that if it's true for n (i.e. "10^(2n-1)+1 divisible by 11"), then it's true for n+1 (i.e. "10^(2(n+1)-1)+1 divisible by 11"). The key point is that

10^(2(n+1)-1)+1 =
10^(2n-1)*100 + 1 =
(10^(2n-1)+1)*100 + 1 - 100

Let it sink in, induction takes a while to grasp at first. Incredible tool though.

Answer 3

11|?[10^(2n-1) + 1]
by induction,
if n=1
10^(2n-1) + 1 = 11
11/11 = 1

if n = 5
10^(2n-1) + 1 = 1000000001
1000000001/11 = 90909091

if n = 6
10^(2n-1) + 1 = 100000000001
100000000001/11 = 9090909091

therefore
10^(2n-1) + 1 is divisible by 11 for all n