Question

For an A.P show that $T_p+T_{p+2q}=2T_{p+q}$

Answer 1

To show that: $T_p+T_{p+2q}=2T_{p+q}$

Proof :-

$T_p=a+(p-1)d$ where a = first term and d = common difference

$T_{p+2q}=a+(p+2q-1)d$

$\therefore T_p+T_{p+2q}=a+(p-1)d+a+(p+2q-1)d=a+pd-d+a+pd+2qd-d=2a+2pd+2qd-2d=2(a+pd+qd-d)=2[a+(p+q)d-d]=2[a+(p+q-1)d]=2T_{p+q}$

Hence proved.