UPSC CDS 2 2020 Math Paper

Concept:The median of an even number of terms in an arithmetic progression is the average of the two middle terms.Explanation:The given series is $2, 4, 6, \ldots, 100$. This is an arithmetic progression with first term $a = 2$, common difference $d = 2$, and last term $l = 100$. Number of terms $n$ is found using $l = a + (n-1)d$: $100 = 2 + (n-1) \times 2 \implies 98 = (n-1) \times 2 \implies 49 = n-1 \implies n = 50$. Since $n = 50$ (even), the median is the average of the $25^{\text{th}}$ and $26^{\text{th}}$ terms. $25^{\text{th}}$ term $= a + (25-1)d = 2 + 24 \times 2 = 2 + 48 = 50$. $26^{\text{th}}$ term $= a + (26-1)d = 2 + 25 \times 2 = 2 + 50 = 52$. Median $= \frac{50 + 52}{2} = \frac{102}{2} = 51$. Alternatively, for any arithmetic progression, the median equals the average of the first and last terms: $\frac{2 + 100}{2} = 51$.Answer:51 (Option D)