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The 4th Term of an A.P. is Three Times the First and the 7th Term Exceeds Twice the Third Term by 1. Find the First Term and the Common Difference. - CBSE Class 10 - Mathematics

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Question

The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.

Solution

In the given problem, let us take the first term as a and the common difference as d

Here, we are given that,

`a_4 = 3a` ......(1)

`a_2 = 2a_3 + 1`  .....(2)

We need to find a and d

So as we know

`a_n = a _ (n -1)d`

For the 4th term (n = 4)

`a_4 = a + (4 -1)d`

`3a = a + 3d`

3a - a = 3d

2a = 3d

`a = 3/2 d`

Similarly for the 3rd term (n = 3)

`a_3 = a + (3 - 1)d`

= a + 2d

Also for the 7th term (n = 7)

`a_7 = a + (7 - 1)d`

= a + 6d  .....(3)

Now, using the value of a3 in equation (2), we get,

`a_7 = 2(a + 2d) + 1`

`= 2a + 4d + 1`   ......(4)

Equating (3) and (4) we get

a + 6d = 2a + 4d +1

6d - 4d - 2a + a= +1

2d - a= +1

`2d - 3/2 d  =1`               `(a = 3/2 d)`

On further simplification, we get,

`(4d - 3d)/2 = 1`

d = (1)(2)

d = 2

Now to find a

`a = 3/2 d

`a =3/2 (2)`

a= 3

Therefore for the given A.P d= 2, a = 3

  Is there an error in this question or solution?

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Solution The 4th Term of an A.P. is Three Times the First and the 7th Term Exceeds Twice the Third Term by 1. Find the First Term and the Common Difference. Concept: nth Term of an AP.
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