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1. The remainder when 3⁰ + 3¹ + 3² + 3³ + . . . . . . . + 3²⁰⁰ is divided by 13 is:

A. 0

B. 4

C. 3

D. 12

Solution:

\begin{aligned} The given expression is in GP series \\ S = 3^{0} + 3^{1} + 3^{2} + 3^{3} + . . . . . . . . + 3^{200} \\ S = \frac{3^{0} \times (3^{201} - 1)}{3 - 1} \\ S = \frac{3^{201} - 1}{2} \\ S = \frac{{(3^{3})}^{67} - 1^{3}}{2} \\ S = \frac{27^{67} - 1^{3}}{2} \end{aligned}

Since, (Aⁿ - Bⁿ) is divisible by (A - B), So, (27⁶⁷ - 1³) is divisible by (27 - 1) = 26
Hence, Expression is also divisible by 13 as it is divisible by 26
Thus given expression is divisible by 13 so the remainder will be 0

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