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Q.
The number of terms common between the two series $2 + 5 + 8 + ...$ up to $50$ terms and $3 + 5 + 7 + 9 + ...$ up to $60$ terms are
Sequences and Series
Solution:
Let $m^{th}$ term of first $A.P$. be equal to the $n^{th}$ term of the second $A.P$. then $2, 5, 8, .... 50$ terms, series $1$
$3, 5, 7, ..., 60$ terms, series $2$
common series $5, 11, 17, ..., 119$
$40^{th}$ term of series $1 = 59^{th}$ term of series $2= 119$ = last term of common series
$\Rightarrow a_{n} = 5 + \left(n - 1\right)d \Rightarrow 119 + 1 = 6n \Rightarrow n = 20$.
$\therefore $ Number of common terms are $20$.