Question

Assertion (A)
Sum of first n positive integers is given by $S_n=\frac{1}{2}n(n+1).$
Reason (R)
In an AP with first term a and common difference d, the sum of nth terms is given by $S_n=\frac{n}{2}\left[2a+(n-1)d\right].$

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R)is true.

Answer 1

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

$$\begin{gathered} S_n=1+2+3+4+...+n \\ =\frac{n}{2}\left[2\times 1+\left(n-1\right)\times 1\right]=\frac{n}{2}\left(n+1\right) \end{gathered}$$
This proves Assertion (A) and clearly, Reason (R) gives Assertion (A).