3^4n+1 + 2^ 2n+2 is divisible by 7
P(n): 34n+1+22n+2 is divisible by 7
step 1: let n=1
34+1+16=243+16=259 is divisible by 7
step 2: let n=k and let P(n) is true for n=k.
34k+1+22k+2=7m
34k+1=7m-22k+2 -(1)
step 3: let n=k+1
34k+5+22k+4
34k+1.34+22k+4
(7m-22k+2)81+22k+4 [ from equation (1) ]
567m-81.22k+2+22k+4
567m-22k+2(81-4)
567m-22k+2(77)
7[ 81m-22k+2(11) ]
P(k+1) is true because P(k) is true
Hence P(n) is true for all natural numbers
step 1: let n=1
34+1+16=243+16=259 is divisible by 7
step 2: let n=k and let P(n) is true for n=k.
34k+1+22k+2=7m
34k+1=7m-22k+2 -(1)
step 3: let n=k+1
34k+5+22k+4
34k+1.34+22k+4
(7m-22k+2)81+22k+4 [ from equation (1) ]
567m-81.22k+2+22k+4
567m-22k+2(81-4)
567m-22k+2(77)
7[ 81m-22k+2(11) ]
P(k+1) is true because P(k) is true
Hence P(n) is true for all natural numbers