Question

The sum of the 4^th and the 8^th terms of an A.P. is 24, and the sum of the 6^th and 10^th term is 34. Find the first term and the common difference of the A.P.

Answer 1

Given : 

The sum of the 4^th and the 8^th terms of an A.P. is 24, and the sum of the 6^th and 10^th term is 34 

⇒ a₄ + a₈ = 24 and 

a₆ + a₁₀ = 34 

We know, 

a_n = a + (n – 1)d

Where a is first term or a₁ and d is the common difference and n is any natural number 

When n = 4 : 

∴ a₄ = a + (4 – 1)d 

⇒ a₄ = a + 3d 

When n = 6 : 

∴ a₆ = a + (6 – 1)d 

⇒ a₆ = a + 5d 

When n = 8 : 

∴ a₈ = a + (8 – 1)d 

⇒ a₈ = a + 7d 

When n = 10 : 

∴ a₁₀ = a + (10 – 1)d 

⇒ a₁₀ = a + 9d 

According to question : 

a₄ + a₈ = 24 

⇒ a + 3d + a + 7d = 24 

⇒ 2a + 10d = 24 

⇒ 2(a + 5d) = 24

⇒ a + 5d = \(\frac{24}{2}\) 

⇒ a + 5d = 12………(i) 

a₆ + a₁₀ = 34 

⇒ a + 5d + a + 9d = 34 

⇒ 2a + 14d = 34 

⇒ 2(a + 7d) = 34 

⇒ a + 7d = \(\frac{34}{2}\)

⇒ a + 7d = 17………(ii) 

Subtracting equation (i) from (ii) : 

a + 7d – (a + 5d) = 17 – 12 

⇒ a + 7d – a – 5d = 5 

⇒ 2d = 5

⇒ d = \(\frac{5}{2}\)

Put the value of d in equation (i) :

Hence, 

The value of a and d are \(-\frac{1}{2}\) and \(\frac{5}{2}\) respectively.