First slide

Introduction to P.M.I

Question

$2^{3 n} - 7 n - 1$ s divisible by

Easy

A

64
B

36
C

49
D

25

Solution

Let $P (n) = 2^{3 n} - 7 n - 1$
$\Rightarrow P (1) = 0, P (2) = 49$
P(1) and P(2) are divisible by 49.
$\begin{matrix} Let P (k) = 2^{3 k} - 7 k - 1 = 49 l \\ P (k + 1) = 2^{3 k + 3} - 7 k - 8 \\ = 8 (49) + 7 k + 1) - 7 k - 8 \\ = 49 (8 /) + 49 k = 49 λ \end{matrix}$
where, $λ = 8 I + k$ which is an integer.

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