Let 3, 6, 9, 12, ....... upto 78 terms and 5, 9, 13, 17, ...... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ________.
Answer (integer)
2223
Solution
<p>1st AP :</p>
<p>3, 6, 9, 12, ....... upto 78 terms</p>
<p>t<sub>78</sub> = 3 + (78 $-$ 1)3</p>
<p>= 3 + 77 $\times$ 3</p>
<p>= 234</p>
<p>2nd AP :</p>
<p>5, 9, 13, 17, ...... upto 59 terms</p>
<p>t<sub>59</sub> = 5 + (59 $-$ 1)4</p>
<p>= 5 + 58 $\times$ 4</p>
<p>= 237</p>
<p>Common term's AP :</p>
<p>First term = 9</p>
<p>Common difference of first AP = 3</p>
<p>And common difference of second AP = 4</p>
<p>$\therefore$ Common difference of common terms</p>
<p>AP = LCM (3, 4) = 12</p>
<p>$\therefore$ New AP = 9, 21, 33, .......</p>
<p>t<sub>n</sub> = 9 + (n $-$ 1)12 $\le$ 234</p>
<p>$\Rightarrow n \le {{237} \over {12}}$</p>
<p>$\Rightarrow n = 19$</p>
<p>$\therefore$ ${S_{19}} = {{19} \over 2}\left[ {2.9 + (19 - 1)12} \right]$</p>
<p>$= 19(9 + 108)$</p>
<p>$= 2223$</p>
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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