Question

In an A P. of even number of terms, the sum of the odd terms is $24$ while that of even terms is $30$ and the last term exceeds the first by $10.5$, then the number of terms is

• A. 8
• B. 6
• C. 10
• D. None of these

Answer 1

The correct option is A 8

Let the number of terms be $2n.1st\text{term}=a,\text{common diff}=d.$

$a_n=a+(2n-1)d=l$

$l-a=10.5,a+(2n-1)d-a=10.5,2nd-d=10.5⟶\left(iii\right)$

For odd terms $24=\frac{n}{2}[2a+(n-1)2d,24=n[a+(n-1)d]⟶(i)$

For even terms $30=\frac{n}{2}\left[2\right(a+d)+(n-1\left)2d\right],30=n\left[\right(a+d)+(n-1\left)d\right]⟶\left(ii\right)$

Subtracting $\left(ii\right)$ and $\left(i\right)$,

$30=an+nd+n(n-1)d$

$24=an$ $+n(n-1)d$

$6=$$nd$

From $\left(iii\right),$

$2nd-d=10.512-d=10.5d=1.5$

$24=n[a+(n-1\left)d\right]$

$24=4[a+(3\left)1.5\right]$

$24=4a+18$

$6=4aa=\frac{6}{4}=1.5$

Number of terms $=2n=8$