Q160 / 270 · NDA 2022 (Phase 2) · Mathematics

Consider the following statements : 1. $2+4+6+\ldots+2 n=n^2+n$ 2. The expression $n^2+n+41$ always gives a prime number for every natural number $n$ Which of the above statements is/are correct?

  • Correct answer
    A
    1 only
  • B
    2 only
  • C
    Both 1 and 2
  • D
    Neither 1 nor 2

Solution

Let $S=2+4+6+\ldots+2 n$ $=2(1+2+3+\ldots+n)=2 \cdot \frac{n(n+1)}{2}=n^2+n$ $\therefore \quad$ Statement 1 is true. Putting $n=41$ in $n^2+n+41$ $=41^2+41+41=41 .(41+2)=41.43$ Which is not prime. So, expression not always prime for $\mathrm{n} \in \mathrm{N}$

Hence, the correct answer is option: 'A'.