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  4. For all n ∈ N, 23n+3 -7n- 8 is divisible by

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Q. For all $n \in N$, $2^{3n+3} -7n- 8$ is divisible by

Binomial Theorem

Solution:

$2^{3n + 3} - 7n- 8 = (2^3)^{n+1} -7(n + 1 ) - 1$
$= \left(1+7\right)^{n+1}-7\left(n+1\right)-1$
$= \,{}^{n+1}C_{0}+ \,{}^{n+1}C_{1}7+\,{}^{n+1}C_{2} 7^{2}+\,{}^{n+1}C_{3}7^{3}+... +\,{}^{n+1}C_{n+1} 7^{n+1}$
$-7\left(n+1\right)-1$
$= 7^{2}\left(\,{}^{n+1}C_{2}+\,{}^{n+1}C_{3}7^{1}+...\,{}^{n+1}C_{n+1}\,7^{n-1}\right)$
which is divisible by $49$.