Question

The number of terms in 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.

Answer 1

$\Rightarrow$Let the total numbers of terms be $2n$

$\therefore$ According to question:

$\Rightarrow$$T_1+T_3+T_5+...+T_{2n-1}=24⟶\left(I\right)$

$\Rightarrow$$T_2+T_4+T_6+...+T_{2n}=30⟶\left(II\right)$

Subtract $\left(I\right)$ from $\left(II\right)$

$\Rightarrow$$(d+d+...)=6$

$\Rightarrow$$nd=6$

$\Rightarrow$$T_{2n}=T_1+\frac{21}{2}$

$\Rightarrow$$T_{2n}-a=\frac{21}{2}$

$⟹a+(2n-1)d-a=\frac{21}{2}$

$⟹2nd-d=\frac{21}{2}$

$⟹2\times 6-d=\frac{21}{2}$

$⟹[d=\frac{3}{2}]$

$\because nd=6$

$\therefore n\times \frac{3}{2}=6$

$⟹n=4$

Total number of terms $=2n=2\times 4=8$