What is the sum of all the common terms between the given series S1 and S2? S1 = 2, 9, 16, . . . . . ., 632 S2 = 7, 11, 15, . . . . . ., 743

Answer: Option C

S₁ = 2, 9, 16, . . . . . ., 632
S₂ = 7, 11, 15, . . . . . ., 743
S₁ = 2, 9, 16, 23, 30, 37, 44, 51 . . . . . .
S₂ = 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, . . . . . .
Number of terms in $${\text{S}}_1=\frac{632-2}{7}+1=91$$
∵ In S₁ there is no further terms beyond 632 therefore there is no common terms.
∴ Number of terms in series S₁ = 91
If this series start from the 23 then number of terms = 91 - 3 = 88
Now, number of common terms in both series S₁ and S₂ = $$\frac{88}{7}$$ = 22 = n
Now, common terms 23, 51, . . . . . . 22
$$\begin{array}{rl}& \text{Sum}=\frac{n}{2}\left[2a+\left(n-1\right)d\right]\\ & =11\left[46+21\times 28\right]\\ & =11\left[46+588\right]\\ & =11\times 634\\ & =6974\end{array}$$