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

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
$$\eqalign{ & {\text{Sum}} = \frac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right] \cr & = 11\left[ {46 + 21 \times 28} \right] \cr & = 11\left[ {46 + 588} \right] \cr & = 11 \times 634 \cr & = 6974 \cr} $$