Question

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

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