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Let $3,6,9,12, \ldots \text { upto } 78$ terms and $5,9,13,17, \ldots \text { upto } 59$ terms be two series. Then the sum of the terms common to both the series is equal to ___________
Option: 1
2223

Option: 2
-

Option: 3
-

Option: 4
-

Answers (1)

best_answer

Both given series are $\mathrm{AP's}$

$3,6,9,12, \ldots \\$

$\mathrm{a=3, d=3, n=78} . \\$

$5,9,13,17, \ldots \\$

$\mathrm{a^{\prime}=5, d^{\prime}=4, n^{\prime}=59}$

Common terms to $\mathrm{2AP's} . \\$ also form an $\mathrm{AP} \\$

First term of common $\mathrm{AP,A=9} \\$

Common Difference $\mathrm{D=lcm(d,d')} \\$

$\mathrm{=\operatorname{lcm}(3,4)}\\ \\$

$\mathrm{=12}$

Let Nth term of this common $\mathrm{AP}$ be its last term

Last term of first $\mathrm{A P =3+(78-1) 3} \\$

$\mathrm{=234}$

Last term of second $\mathrm{A P =5+(59-1) 4} \\$

$\mathrm{=237}$

$\mathrm{\therefore \quad A+(N-1) D \leqslant 234} \\$

$\mathrm{9+(N-1) 12 \leqslant 234} \\$

$\mathrm{N-1 \leq \frac{225}{12}} \\$

$\mathrm{\Rightarrow \quad N \leqslant 19.75} \\$

$\mathrm{\therefore \quad N=19}$

$\mathrm{\text{Sum of common AP}}\\$

$\mathrm{=\frac{19}{2}[2 \cdot 9+(19-1) 12]} \\$

$\mathrm{=19[9+18*6]} \\$

$\mathrm{=2223} \\$

Hence answer is $\mathrm{2223}$

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