Question

Prove 5^(2n+2)-24^n-25 is divisible by 576

Answer 1

5^(2n+2) - 24n - 25 

is divisivle by 576

If n = 1,
f(n) = 5^4 - 24 - 25 = 625 - 49 = 576.

Assume true for n = r
f(r+1) = 5^(2r + 4) - 24r - 24 - 25
= 25.5^(2r+2) - 24r - 25 - 24
= 25.5^(2r+2) - 25.24r - 25.25 - 24 + 24.24.r + 24.25
= 25.f(r) - 576r + 576
Each term divisible by 576 so whole expression is.

Answer 2

Step-by-step explanation:

step 1:

for n=1

p(1):5^4-24-25=625-49=576,which is divisible by 576.

so p(1) is true.

step 2:

Assume that,p(n) is true for all k.

p(k):5^2k+2-24k-25 is divisible by 576.

step 3:

Fpr n=k+1

To prove that p(k+1) is true.

i.e to prove that,p(k+1):5^2(k+1)+2-24(k+1)-25 is divisible by 57

=5^2k+2×5^2-24k-24-25

from equation 1,

=(576m+24k+25)×5^2-24k-49

=25×576m+600k+625-24k-49

=25×576m+576k+576

=576(25m+k+1),which is divisible by 576.

so p(k+1)is true for all n belongs to N.