Question

In an AP :
Given $a_3=15,S_{10}=125$ find $d$ and $a_{10}$

Answer 1

$a_3=15$

Let the first term $=a$

Common difference $=d$

$\therefore a_3=a+2d=15⟶\left(1\right)$

$S_{10}=\frac{10}{2}[2a+9d]=125$

$2a+9d=25⟶\left(2\right)$

multiplying $\left(1\right)$ by $2$ and then subtracting with $\left(2\right)$

$2a+4d=30$

$2a+9d=25$

$---$

$\overline{-5d=5}$

$\Rightarrow d=-1$

putting $d$ in equation $\left(1\right)$

$\Rightarrow a+(-1)2=15$

$\Rightarrow a=17$

$\Rightarrow a_{10}=a+9d=17-7=8$

Hence, the answer is $-1,8.$