Question

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

Answer 1

Let total no. of terms be $2n$, & first term, $a$ and comon diff be $d$.

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

$l-a=10.5$

if we take odd terms only it will also be a $A.P$. of n terms only it will also be a $A.P$ of n terms with common diff. of $2d$.

$24=n/2\{2a+(n-1\left)2d\right\}$

$24=n(a+(n-1\left)d\right)$..........(2)

$30=n/2\left\{2\right(a+d)+(n-1\left)2d\right\}$

$30=n(a+nd)$ .............. (3)

Put the value of equation (3) in equation (2)

$24=30-nd$

$nd=6$, putting in equation (1)

$12-d=10.5$

$d=1.5$ so $n=4$

$a=1.5$

so no of terms $8$.

series $1.5,3,4.5,6,7.5,9,10.5,12$