Question

If a sequence whose $n^{th}$ term is given by
$t_n$ = 3.23 $n^2$– 4.81 n + 37, then which of the following is true?

• A. The above sequence is an AP only for some integers.
• B. The above sequence is not an AP
• C. The above sequence is an AP for all integers.
• D. The above sequence is an AP for all negative integers.

Answer 1

The correct option is B

The above sequence is not an AP

In an AP, the common difference is a constant, i.e,$t_n$ – $t_{n-1}$ must not contain any variables, it should be a constant number.

$t_n$ – $t_{n-1}$ = (3.23$n^2$ – 4.81n + 37) – (3.23${(n-1)}^2$ – 4.81(n-1) + 37)

Observe that the coefficient of $n^2$ will be zero because it is multiplied with 3.23 in both $t_n$ and $t_{n-1}$. The -4.81n in the first braces and the -4.81n from the$2^{nd}$ braces will cancel off. However we shall still be left over with a non-zero coefficient of n from the $2^{nd}$ braces from the expansion of ${(n-1)}^2$, i.e. the common difference depends on ‘n’ and is not constant. Hence this sequence is not an AP.