Write the two arithmetic progressions

• First series: 3,6,9,12,, ext{ upto 78 terms} This is an AP with d_1=3$$ So its terms are $$3n ext{ for } n=1,2,,78$$ Hence the last term is $$3\times 78=234.$$
• Second series: 5,9,13,17,, ext{ upto 59 terms} This is an AP with d_2=4$$ Its general term is $$5+(n-1)4=4n+1,  n=1,2,,59.$$ The last term is $$5+(59-1)4=5+232=237.$$

Find the common terms

A common term must be:

• a multiple of 3, and
• of the form $5+4k$ for some integer $k\ge 0$.

So solve

Find the last common term within both series

The common terms must not exceed the smaller of the two last terms: $\min(234,237)=234.$

So the common-term AP is 9,21,33,,233? Since terms increase by 12, check the largest term of the form $9+12m\le 234.$

Solve:

Find the number of common terms

For the AP 9,21,33,,225 with first term $9$, common difference $12$, last term $225$:

$n=\frac{225-9}{12}+1=\frac{216}{12}+1=18+1=19.$

Find the sum of common terms

Sum of $19$ terms of this AP: $S=\frac{n}{2}(a+l)$ $S=\frac{19}{2}(9+225)$ $S=\frac{19}{2}\cdot 234$ $S=19\cdot 117=2223.$

Compare with stored answer

Our derived answer is $2223$ which matches the stored correct answer.