The 8^{text{th}} common term of the series S_ | vedclass

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Question

(B) The series $S_1$ is an arithmetic progression with first term $a_1 = 3$ and common difference $d_1 = 4$. Its general term is $T_n = 3 + (n-1)4 = 4

Vedclass · Accepted Answer

$151$

Vedclass · Answer

(B) The series $S_1$ is an arithmetic progression with first term $a_1 = 3$ and common difference $d_1 = 4$. Its general term is $T_n = 3 + (n-1)4 = 4n - 1$.The series $S_2$ is an arithmetic progression with first term $a_2 = 1$ and common difference $d_2 = 5$. Its general term is $T_m = 1 + (m-1)5 = 5m - 4$.For a common term,$4n - 1 = 5m - 4$,which implies $4n + 3 = 5m$.The first common term is $11$ (for $n=3, m=3$).The common difference of the new series formed by common terms is the least common multiple of $d_1$ and $d_2$,which is $	ext{lcm}(4, 5) = 20$.The $k^{	ext{th}}$ common term is given by $T_k = 11 + (k-1)20$.For $k=8$,$T_8 = 11 + (8-1) 	imes 20 = 11 + 7 	imes 20 = 11 + 140 = 151$.

The $8^{\text{th}}$ common term of the series $S_1 = 3 + 7 + 11 + 15 + 19 + \dots$ and $S_2 = 1 + 6 + 11 + 16 + 21 + \dots$ is $.......$.

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