UPSC CDS 1 2020 Math Paper

Concept:The expression can be factored as $3^x \times 13$. The largest divisor from the options that divides this expression for every natural number $x$ is to be found.Explanation:Factor the given expression:$3^x + 3^{x+1} + 3^{x+2} = 3^x(1 + 3 + 3^2) = 3^x \times 13$.For any natural number $x$, the expression contains at least one factor of $3$ and one factor of $13$.The smallest value occurs when $x = 1$, giving $3^1 \times 13 = 39$.Thus, the expression is always a multiple of $39$.Check the options:• $3$ always divides the expression (since $3^x$ is a factor).• $13$ always divides the expression.• $39 = 3 \times 13$ also always divides the expression.• $117 = 3^2 \times 13$ divides the expression only when $x \ge 2$, but not when $x = 1$ (since $39$ is not a multiple of $117$).Therefore, the largest divisor that works for every natural number $x$ is $39$.Answer:$39$ (Option C).