Question

The number of terms of an $A.P$. is even; the sum of the odd terms is $24$ , sum of the even terms is $30$ , and the last term exceeds the first by $10\frac{1}{2}$, then the number of terms in the series is

• A. 3
• B. 4
• C. 6
• D. 5

Answer 1

Answer: (B)

Let the series has $2n$ terms and the series is
$a,a+d,a+2d,\dots ,a+(2n-1)d$
According to the given conditions
We have
$[a+(a+2d)+(a+4d)+\dots (a+(2n-2)d)]=24$
$\Rightarrow \frac{n}{2}[2a+(n-1)2d]=24$
$\Rightarrow n[a+(n-1)d]=24$ ...(1)
Also $[(a+d)+(a+3d)+\dots (a+(2n-1)d)]=30$
$\Rightarrow \frac{n}{2}[2(a+d)+(n-1)2d]=30$
$\Rightarrow n[(a+d)+(n-1)d]=30$ ...(2)
Also last term exceeds the first by $21/2$
$\Rightarrow a+(2n-1)d-a=21/2$
$\Rightarrow (2n-1)d=21/2$ ...(3)
Now subtracting (1) from (2)
$nd=6$ ...(4)
Dividing $(3)$ by (4)
$\Rightarrow \frac{2n-1}{n}=\frac{21}{12}$
$\Rightarrow n=4$