Question

Assertion :(A): $S_n=n^3+3n^2+5n+3$, then $t_n$ is divisible by $3$. Reason: (R): $t_n=3\lambda$ , where $\lambda$ is an integer.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).

$S_u=n^3+3n^2+5n+3$
$\therefore t_n=S_u-S_{n-1}$
$=(n^3+3n^2+5n+3)-\left\{\right((n-1{)}^3+3(n-1{)}^2+5(n-1)+3\}$
$=3n^2+3n+3=3(n^2+n+1)=3\lambda$
Therefore, $t_n$ is divisible by $3$.