# 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 'a_{1}' 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 }= a_{1} + (n - 1) d

Taking n = 6,

⇒ a_{6 }= a_{1} + (6 - 1) d

⇒ a_{6 }= a_{1} + 5d ----------------- (1)

Given that, a_{6 }= 4a_{1 }------------------ (2)

Substituting (2) in (1) we get,

⇒ 4a_{1} = a_{1} + 5d

⇒ 3a_{1 }= 5d

⇒ a_{1 }= 5d/3 -------------- (3)

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

S_{n }= n / 2 [2a_{1} + ( n - 1) d ]

Given that, S_{6 }= 75 and using a_{1 }= 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,

a_{1 }= (5 × 3) /3

a_{1 }= 5

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

visual curriculum