UPSC NDA I 2021 Mathematics Solved Paper

Concept:A geometric progression (GP) is defined by a fixed common ratio between consecutive terms.Multiplying or dividing every term by the same non‑zero number does not change this common ratio.Explanation:Let the original GP be $a_1, a_2, a_3, \dots, a_n$ with common ratio $r$.So $r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \dots = \frac{a_n}{a_{n-1}}$.If each term is multiplied by a non‑zero constant $k$, the new sequence becomes $ka_1, ka_2, ka_3, \dots, ka_n$.Its common ratio is $\frac{ka_2}{ka_1} = \frac{a_2}{a_1} = r$. The ratio remains the same.Similarly, if each term is divided by the same non‑zero $k$, the new terms are $\frac{a_1}{k}, \frac{a_2}{k}, \frac{a_3}{k}, \dots$.The common ratio is $\frac{a_2/k}{a_1/k} = \frac{a_2}{a_1} = r$. Again, the ratio is unchanged.Therefore, both operations preserve the defining property of a GP — a constant common ratio.Hence, both statements 1 and 2 are correct.Answer:Both 1 and 2 are correct. (Option C)