Question

Which of the following is/are true?
I. The sum of first $23$ terms of the A.P. $16,11,6, \ldots .$. is $-897$.
II. The sum of first $22$ terms of the A.P. $x+y, x-y, x-3 y, \ldots$. is $22[x-20 y]$.

• 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)

I. Given A.P. is $16,11,6, \ldots$
Here, $a=16, d=11-16=-5$
$\because S_n=\frac{n}{2}[2 a+(n-1) d] $
$\therefore S_{23}=\frac{23}{2}[2 \times 16+(23-1)(-5)]$
$ =\frac{23}{2}[32+(22)(-5)]=\frac{23}{2}[32-110] $
$=\frac{23}{2}[-78]=-897 $
II. Given A.P. is $x+y, x-y, x-3 y, \ldots$
Here, $ a=x+y$
$d=(x-y)-(x+y)=-2 y $
$\because S_n=\frac{n}{2}[2 a+(n-1) d]$
$\therefore S_{22}=\frac{22}{2}[2 \times(x+y)+(22-1)(-2 y)] $
$ =11[2 x+2 y+(21)(-2 y)]=11[2 x+2 y-42 y] $
$=11[2 x-40 y]$
$=22[x-20 y] $