Prove that 11^{n+2 }+ 12^{2n+1 }is divisible by 133 for all n belongs to N .

for n =1 we get

3059/133 = 23

hence it is true for n =1

let us assume that this expression is true for n = k

is divisible by 133

for n = k+1

=

=

which is divisible by 1331+1728 = 3059 which is divisible by 133

hence our assumption is correct and is divisible by 133

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