Find the sum of the 100 common terms of the given two series : 10, 16, 22, ...…

Answer: 30700

Explanation:

• First sequence: 10, 16, 22, ... (arithmetic progression with first term 10 and difference 6).

• Second sequence: 10, 13, 16, 19, 22, ... (arithmetic progression with first term 10 and difference 3).

• Every term of the first sequence appears in the second sequence because the first sequence increases by 6, which is a multiple of 3, so its terms share the same residue modulo 3 as the second sequence. Therefore the common terms are exactly 10, 16, 22, ... .

• We need the sum of the first 100 terms of this AP: a = 10, d = 6, n = 100.

• Use the formula S_n = (n/2)[2a + (n-1)d].

• Compute: S_100 = (100/2)[2*10 + 99*6] = 50[20 + 594] = 50*614 = 30700.