Question : The remainder of the term $9+9^2+\ldots+9^{(2 n+1)}$ when divided by 6 is:
Option 1: 1
Option 2: 4
Option 3: 2
Option 4: 3
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Correct Answer: 3
Solution :
Given, $9+9^2+\ldots+9^{(2 n+1)}$ is divided by 6
Consider, $9+9^2+\ldots+9^{(2 n+1)}$
Here, for each term in the series, the remainder when divided by 6 is 3.
The sum of the remainder of each term = 3 + 3 + 3 + ............+ up to 2n + 1 terms
2n + 1 is an odd number, so when the sum of an odd number times 3 is divided by 6, the remainder is 3.
Hence, the correct answer is 3.
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