The $8^{\text {th }}$ common term of the series $S_{1}=3+7+11+15+19+\ldots .$, $S_{2}=1+6+11+16+21+\ldots$. is \_\_\_\_ .

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Sequences and Series

The

8^{th}

common term of the series

S_{1} = 3 + 7 + 11 + 15 + 19 + \dots .

S_{2} = 1 + 6 + 11 + 16 + 21 + \dots

.
is ____ .

Answer: 151

The general terms are

4 n - 1

for

S_{1}

and

5 m - 4

for

S_{2}

. A common term needs

4 n - 1 = 5 m - 4

, whose first solution is

11

. Because the two series step by

4

and

5

, their common terms form an A.P. with first term

11

and common difference

lcm (4, 5) = 20

. The eighth common term is

11 + (8 - 1) \times 20 = 151

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Integer answer