Let 3,6,9,12, up to 78 terms and 5,9,13,17, up to 59 terms be two series.

Question

Let $3,6,9,12,\ldots$ up to $78$ terms and $5,9,13,17,\ldots$ up to $59$ terms be two series. Then the sum of the terms common to both series is equal to \_\_\_\_\_\_\_\_.

Answer: 2223

Accepted Answer

Last terms: $t_{78}=234$ (first AP), $t_{59}=237$ (second AP).

Common terms form an A.P. with first term $9$ and common difference $\mathrm{lcm}(3,4)=12$: $9,21,33,\ldots \le234$, giving $n\le\dfrac{237}{12} \Rightarrow n=19$.

$$S_{19} = \frac{19}{2}\big[2(9)+18(12)\big] = 19(9+108) = 2223.$$

Solutions · 1