7ⁿ-3ⁿ is a divisible by 4. 

please solve this

it can be prvd byP.M.I.

for n=1

7ⁿ-3ⁿ=7-3=4 which is div. by 4

so true fr n=1

now,

let it be true for n=k

then,

7^k-3^k=4m 

7^k=4m+3^k...........(1)

now, for n=k+1

prove it for n=k

pt. 7^k+1-3^k+1 is divisible by 4

now,7^k+1-3^k+1=7.7^k-3^k+1

=7(4m+3^k)-3^k+1.........from eq.1

=7.4m+7.3^k-3.3^k

=7.4m+3^k(7-3)

=7.4m+3^k.4

=4(7m+3^k)+0

since remainder =0

so it is divisible by 4

so tru for n=k+1

so

by p.m.i. it is tru for any valueof(n=natural no.)