**Question:**

Find the number of terms common to the two arithmetic progressions 5, 9, 13, 17, …., 217 and 3, 9, 15, 21, …., 321.

**Solution:**

To Find: The number of terms common to both AP

Given: The 2 AP’s are 5, 9, 13, 17, …., 217 and 3, 9, 15, 21, …., 321

As we find that first common term of both AP is 9 and the second common term of both AP is 21

Let suppose the new AP whose first term is 9, the second term is 21, and the common difference is 21 – 9 = 12

NOTE: As first AP the last term is 217 and second AP last term is 321. So last term of supposing AP should be less than or equal to 217 because after that there are no common terms

Formula Used: $T_{n}=a+(n-1) d$

(Where $T_{n}$ is nth term and $d$ is common difference of given AP)

$217 \geq a+(n-1) d \Rightarrow 9+(n-1) 12 \leq 217$

$\therefore(n-1) 12 \leq 208 \Rightarrow(n-1) \leq 17.33 \Rightarrow n \leq 18.33$

So, Number of terms common to both AP is 18.