# 3^4n+1 + 2^ 2n+2 is divisible by 7

^{4n}

^{+1}+2

^{2n}

^{+2}is divisible by 7

step 1: let n=1

3

^{4}

^{+1}+16=243+16=259 is divisible by 7

step 2: let n=k and let P(n) is true for n=k.

3

^{4k}

^{+1}+2

^{2k+2}=7m

3

^{4k}

^{+1}=7m-2

^{2k+2}-(1)

step 3: let n=k+1

3

^{4k+5}+2

^{2k+4}

3

^{4k+1}.3

^{4}+2

^{2k+4}

(7m-2

^{2k+2})81+2

^{2k+4}[ from equation (1) ]

567m-81.2

^{2k+2}+2

^{2k+4}

567m-2

^{2k+2}(81-4)

567m-2

^{2k+2}(77)

7[ 81m-2

^{2k+2}(11) ]

P(k+1) is true because P(k) is true

Hence P(n) is true for all natural numbers