First slide

Geometric progression

Question

The largest value of the positive integer k for which $n^{k} + 1$ divides $1 + n + n^{2} + \dots + n^{127}$ is divisible by

Moderate

A

8
B

16
C

32
D

64

Solution

We have,
$\begin{array}{l} 1 + n + n^{2} + \dots + n^{127} = \frac{n^{128} - 1}{n - 1} \\ = \frac{(n^{64} - 1) (n^{64} + 1)}{n - 1} \\ = (1 + n + n^{2} + \dots + n^{63}) (n^{64} + 1) \end{array}$
$∴$ k = 64 which is divisible by 8, 16, 32 and 64.

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