# 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.

Given that:

The 6th term of an AP is equal to four times its first term and the sum of first six terms is 75

Let us assume first term = a

Common difference = d

Number of terms = n

According to given condition:

a6 = 4a ———- (1)

We know that a 6 =a + 5d ———-(2)

Substituting eq (2) in (1) we get

a + 5d =4a

3a = 5d

a= 5d / 3 ————-(3)

Sn= n / 2 X [2a + (n-1) d]

So substituting the given data and values from above equation we get

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

75= 3[ 10d/3 + 5d]

75=3(25d/3)

75= 25d

d=3

Substituting the value of d in eq (3)

We get,

a= 5d / 3

a= 5(3)/3

a=5

Therefore, the first term a = 5 and d = 3

Given: The 6th term of an AP is equal to four times its first term and the sum of first six terms is 75

Let a is the first term and d is the common difference and n is the number of terms

Now

Now

a= 5 and d= 3