First slide

Introduction to P.M.I

Question

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

Moderate

A

64
B

36
C

49
D

25

Solution

Let
$\begin{array}{l} P (n) = 2^{3 n} - 7 n - 1 \\ P (1) = 0, P (2) = 49 \end{array}$
P(1) and P(2) are divisible by 49.
$P (k) \equiv 2^{3 k} - 7 k - 1 = 49$ m
$\begin{matrix} ∴ P (k + 1) \equiv 2^{3 k + 3} - 7 k - 8 = 8 (49 m + 7 k + 1) - 7 k - 8 \\ = 49 (8 m) + 49 k = 49 p \end{matrix}$
Hence, by mathematical induction $2^{3 n} - 7 n - 1$ is
divisible by 49.

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