The number of terms common between the two se | vedclass

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(B) The first series is an Arithmetic Progression $(AP)$ with $a_1 = 2$,$d_1 = 3$,and $n_1 = 50$. The $50^{th}$ term is $T_{50} = 2 + (50 - 1)3 = 2 +

Vedclass · Accepted Answer

$20$

Vedclass · Answer

(B) The first series is an Arithmetic Progression $(AP)$ with $a_1 = 2$,$d_1 = 3$,and $n_1 = 50$. The $50^{th}$ term is $T_{50} = 2 + (50 - 1)3 = 2 + 147 = 149$.The second series is an $AP$ with $a_2 = 3$,$d_2 = 2$,and $n_2 = 60$. The $60^{th}$ term is $T'_{60} = 3 + (60 - 1)2 = 3 + 118 = 121$.Common terms must satisfy both sequences. The first few common terms are $5, 11, 17, \dots$. This forms a new $AP$ with first term $A = 5$ and common difference $D = 	ext{lcm}(3, 2) = 6$.The general term of the common series is $T_n = 5 + (n - 1)6 = 6n - 1$.Since the common terms must be present in both series,$T_n \leq 121$ (the smaller of the two last terms).$6n - 1 \leq 121 \implies 6n \leq 122 \implies n \leq 20.33$.Since $n$ must be an integer,the number of common terms is $20$.

The number of terms common between the two series $2 + 5 + 8 + \dots$ up to $50$ terms and the series $3 + 5 + 7 + 9 + \dots$ up to $60$ terms is:

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