Question

If the sequence is $54,51,48, \ldots$. , 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 $54,51,48, \ldots$
Here, $ a=54, d=51-54=-3$
Let $n$ terms are required to give the sum $513$ .
$ S_n=513$
$\Rightarrow \frac{n}{2}[2 a+(n-1) d]=513$
$ \Rightarrow \frac{n}{2}[2 \times 54+(n-1)(-3)]=513$
$ \Rightarrow n(108-3 n+3)=513 \times 2$
$ \Rightarrow -3 n^2+111 n=1026$
$ \Rightarrow -\left(3 n^2-111 n+1026\right)=0$
$ \Rightarrow n^2-37 n+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 $513$.