# Given A = 1,8,15,…1975. B = 2,13,24…1982. Find the number of terms common to both the arithmetic progressions.

Given, A = 1, 8, 15, …. 1975. So, d1 = 7

B = 2, 13, 24, ….1982. So, d2 = 11

Now, d(common) = LCM(d1, d2) = LCM (7, 11) = 77

Here, the first common term will be 57

In order to find the number of common terms between the two series, we write the inequality

57 + (n – 1)77 < 1975

(n – 1)77 < 1918

(n – 1)<(1918/77)

(n – 1)<24.9

n<25.9

n=25

57+(n – 1)77 < 1982

(n – 1)77 < 1925

(n – 1) < 25

n < 26

n = 25

So, the first common term of the series is 57 and the last one is 57 + (25 – 1)77

= 57 + 24×77

=1905