If the 6th term of an A.P. is equal to four times its first term and the sum of first six term is 75 ,find the first term and common difference.

A series of numbers that follow a specific pattern is called an Arithmetic Progression.

Answer: If the 6th term of an A.P. is equal to four times its first term and the sum of first six terms is 75 then, the first term is 5 and the common difference is 3.

Let's solve step by step.

Explanation:

Let the first term of the series be 'a₁' and the common difference be 'd'.

As we know that the formula to find n^th term of a progression is given by,

a_n = a₁ + (n - 1) d

Taking n = 6,

⇒ a₆ = a₁ + (6 - 1) d

⇒ a₆ = a₁ + 5d ----------------- (1)

Given that, a₆ = 4a₁ ------------------ (2)

Substituting (2) in (1) we get,

⇒ 4a₁ = a₁ + 5d

⇒ 3a₁ = 5d  

⇒ a₁ = 5d/3 -------------- (3)

Sum of n^th term of arithmetic progression is given by,

S_n = n / 2 [2a₁ + ( n - 1) d ] 

Given that, S₆ = 75 and using  a₁   = 5d/3 

⇒ 75 = 6 / 2 [2 (5d/3) + ( 6 - 1) d ]   

⇒ 75 = 3  [ (10d/3) + 5d ]   

⇒ 75 = 3 (10d + 15d ) / 3

⇒ 75 = 10d + 15d

⇒ 75 = 25d

⇒ d = 75 / 25

⇒ d = 3

Putting the value of d in (3) we get,

a₁ = (5 × 3) /3

a₁ = 5

Thus, the first term is 5 and the common difference is 3.