Question

If the sequence is $54,51,48,\dots$. , then the sum of first
I. 18 terms is 513.
II. 19 terms is 513 .

• A. Both I and II are true
• B. Only I is true
• C. Only II is true
• D. Both I and II are false

Answer 1

Answer: (A)

Given sequence is $54,51,48,\dots$
Here, $a=54,d=51-54=-3$
Let $n$ terms are required to give the sum $513$ .
$S_n=513$
$\Rightarrow \frac{n}{2}[2a+(n-1)d]=513$
$\Rightarrow \frac{n}{2}[2\times 54+(n-1)(-3)]=513$
$\Rightarrow n(108-3n+3)=513\times 2$
$\Rightarrow -3n^2+111n=1026$
$\Rightarrow -\left(3n^2-111n+1026\right)=0$
$\Rightarrow n^2-37n+342=0$
$\Rightarrow (n-18)(n-19)=0$
$\Rightarrow n=18\text{ or }=19$
Here, the common difference is negative and $19^{\text{th }}$ term
$=54+(19-1)\times (-3)=0$
So, the sum of $18$ terms as well as that of $19$ terms is $513$.