The sum of the first three terms of an Arithmetic Progression (A.P) is 42 and the product of the first and third term is 52.Find the first term and the common difference.

Let the three consecutive terms of the A.P. be :

a - d, a, a + d

Given,

The sum of the first three terms is 42.

⇒ a - d + a + a + d = 42

⇒ 3a = 42

⇒ a = $\dfrac{42}{3}$

⇒ a = 14.

Given,

The product of the first term and the third term is 52:

⇒ (a - d)(a + d) = 52

⇒ (14 - d)(14 + d) = 52

⇒ (14)² - d² = 52

⇒ 196 - d² = 52

⇒ 196 - 52 = d²

⇒ 144 = d²

⇒ d² = 144

⇒ d = $\sqrt{144}$

⇒ d = 12 or -12

Case 1: a = 14 and d = 12

First term = a - d = 14 - 12 = 2

Case 2: a = 14 and d = -12

First term = a - d = 14 - (-12) = 26

Hence, first term = 2 and common difference = 12 or first term = 26 and common difference = -12.