Question

If sum of 3^rd and 8^th terms of an A.P. is 7 and sum of 7^th and 14^th terms is –3 then find the 10 th term .

Answer 1

It is given that,
a₃ + a₈ = 7
a₇ + a₁₄ = –3

Now,
$$\begin{gathered} a_n=a+\left(n-1\right)d \\ {a}_3+a_8=7 \\ \Rightarrow \left(a+\left(3-1\right)d\right)+\left(a+\left(8-1\right)d\right)=7 \\ \Rightarrow a+2d+a+7d=7 \\ \Rightarrow 2a+9d=7 \\ \Rightarrow 2a=7-9d...\left(1\right) \\ {a}_7+a_{14}=-3 \\ \Rightarrow \left(a+\left(7-1\right)d\right)+\left(a+\left(14-1\right)d\right)=-3 \\ \Rightarrow a+6d+a+13d=-3 \\ \Rightarrow 2a+19d=-3 \\ \Rightarrow 7-9d+19d=-3\left(\mathrm{from}\left(1\right)\right) \\ \Rightarrow 7+10d=-3 \\ \Rightarrow 10d=-3-7 \\ \Rightarrow 10d=-10 \\ \Rightarrow d=-1 \\ \Rightarrow 2a=7-9\left(-1\right)\left(\mathrm{from}\left(1\right)\right) \\ \Rightarrow 2a=7+9 \\ \Rightarrow 2a=16 \\ \Rightarrow a=8 \end{gathered}$$

Now,
$$\begin{gathered} a_{10}=a+\left(10-1\right)d \\ =8+9\left(-1\right) \\ =8-9 \\ =-1 \end{gathered}$$

Hence, the 10^th term is –1.