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: Tn = a + (n – 1)d
(Where Tn is nth term and d is common difference of given AP)
217 ≥ a + (n – 1)d ⇒ 9 + (n – 1)12 ≤ 217
(n – 1)12 ≤ 208 ⇒ (n – 1) ≤ 17.33 ⇒ n ≤ 18.33
So, Number of terms common to both AP is 18.