First slide

Geometric progression

Question

The greatest positive integer k, for which $49^{k} + 1$ is a factor of the sum $49^{125} + 49^{124} + ...... + 49^{2} + 49 + 1,$ is

Easy

A

65
B

32
C

60
D

63

Solution

${(49)}^{125} + {(49)}^{124} + . . . . + 49 + 1 = \frac{{(49)}^{126} - 1}{48} = \frac{({(49)}^{63} + 1) ({(49)}^{63} - 1)}{48} \Rightarrow {(49)}^{63} + 1 i s a f a c t o r o f t h e t h e s u m . \Rightarrow k = 63$

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