Prove by mathematical induction that, for all non-negative integers n, 11^(2n) + 25^n + 22 is divisible by 24

The procedure of mathematical induction is as following:

Firstly prove the base case is true i.e. when n=1, the statement is true.

Then assume for some integer n=k the statement is true, and then prove the case n=k+1, the statement is true.

Make a conclusion that by mathematical induction, the statement is true.

For this particular question, the base case is when n=0, the statement is true, since it is asked for 'all non-negative integers'. It is because 11^2x0+25⁰+22=24 is divisible by 24 (24=24x1).

Then let's say P(n) is the proposition that 11²ⁿ + 25ⁿ + 22 is divisible by 24. Assuming that P(k) is true for some integer k=n, then 11^2k + 25^k+ 22  is divisible by 24.

The most important step comes: we then prove that P(k+1) is true. i.e. 11^2k+2 + 25^k+1 + 22 is divisible by 24.

It is true because 11^2k+2 + 25^k+1 + 22=121x 11^2k+25x25^k+22=(120 + 1)11^2k + (24 + 1)25^k + 22= (120 x11^2k + 24 x25^k​)+ (11^2k + 25^k+ 22). Expressions in both brackets are divisible by 24. so P(k+1) is true.

Then we are done. We could conclude that by mathematical induction the statement is true for all non-negative integers.