Using PMI, prove that 

5²ⁿ⁺²-24n-25 is divisible by 576 for n belongs to N.         

 let p (n ) = 5²ⁿ⁺²-24n-25

p(1) = 5⁴ - 24 -25 = 576 -------- true

assume p(k) = 5^2k+2 - 24k - 25 is true 

=> 5^2k+2 - 24k - 25 = 576 x c   where c is some constant

=> 5^2k x 5² = 576c + 24k + 25 ------------ 1

to prove : p ( k+1) is true ie... 5^2k+4 - 24 - 24k - 25 is div by 576

5^2k+4 - 24 - 24k - 25  =  5^2k x 5² x 5² - 24k -49

 = ( 576c + 24k + 25 )5² - 24k - 49          since 5^2k x 5² = 576c + 24k + 25

=  576 x 25 c + 25 x 24 k + 625 -24k - 49

= 576 x 25c + 24k ( 25-1) +576

= 576 x 25c + 24k x 24 + 576

= 576 ( 25c + k + 1 )         which is a multiple of 576 

therefore p( k+1 ) is true

by MI 5²ⁿ⁺²-24n-25 is divisible by 576 for n belongs to N.