UPSC CDS I 2019 Math Paper

Concept:Consecutive natural numbers form an arithmetic progression with common difference $1$.When $n$ new consecutive numbers are added, the average increases by $\frac{n}{2}$.Explanation:The average of $50$ consecutive natural numbers is given as $x$.Let the numbers be $a, a+1, a+2, \dots, a+49$.Their average is $a + \frac{49}{2}$, which equals $x$.Now, the next four numbers are $a+50, a+51, a+52, a+53$.The total of the new set of $54$ numbers equals the original sum plus the sum of the four new numbers.The sum of the four new numbers = $(a+50)+(a+51)+(a+52)+(a+53) = 4a + 206$.Alternatively, using the property of AP: when $n$ new consecutive terms are added, the average increases by $\frac{n}{2}$.Here $n = 4$, so increase = $\frac{4}{2} = 2$.Thus, the new average = $x + 2$.Verification with an example: Let numbers be $1$ to $50$; average $x = \frac{51}{2}$.New numbers $51$ to $54$; total $= \frac{54 \times 55}{2} = 1485$; new average $= \frac{1485}{54} = \frac{55}{2} = \frac{51}{2} + 2 = x+2$.Answer:$x + 2$ (Option B).