A question regarding the common terms of two APs.

Question

What is the sum of first $50$ terms common to the $AP$ $15,19,23,\dots$ and the $AP$ $14,19,24,\dots$? I know that: The common terms start from $19$ and nothing else. I have tried this but I am facing a lot of difficulty. Please help me.

Answer 1

Consider that $(4\mathbf{Z}+15) \cap (5\mathbf{Z}+14)=20\mathbf{Z}+19$. Hence $$ \sum_{i=0}^{49}20n+19=20\left(\sum_{i=0}^{49}n\right)+19\cdot 50=20\cdot \frac{49\cdot 50}{2}+19\cdot 50. $$

Answer 2

We have:

• $a_n=15+\color\red4n$
• $b_n=14+\color\green5n$

We know that:

• $LCM(\color\red4,\color\green5)=20$
• The first common element is $19$

Hence the AP of common elements is $c_n=19+20n$.

And the sum of the first $50$ elements is $50(c_{0}+c_{49})/2=25450$.