Question

If the fifth term of a AP is 19 and the tenth term is 34 Find:- a) the first term b) The common difference c) The sum of the first 15 term

Answer 1

a) The first term is a=7

b) The common difference is d=3

c) The sum of the first 15 term is 420.

Explanation:

Given : If the fifth term of a AP is 19 and the tenth term is 34.

To find : a) the first term b) The common difference c) The sum of the first 15 term ?

Solution :

The Arithmetic progression is in the form,
`a,a+d,a+2d,a+3d,...`

Where, a is the first term and d is the common difference

The nth term of the A.P is
`a_n=a+(n-1)d`

The fifth term of a AP is 19.

`a_5=a+(5-1)d`

`19=a+4d` ...(1)

The tenth term is 34.

`a_(10)=a+(10-1)d`

`34=a+9d` ...(2)

Solving (1) and (2) by subtracting the equations,

`34-19=(a+9d)-(a+4d)`

`15=a+9d-a-4d`

`15=5d`

`d=3`

Substitute in (1),

`19=a+4(3)`

`a=19-12`

`a=7`

a) The first term is a=7

b) The common difference is d=3

c) The sum of the first 15 term is given by,
`S_n=(n)/(2)[2a+(n-1)d]`

`S_(15)=(15)/(2)[2(7)+(15-1)3]`

`S_(15)=(15)/(2)[14+(14)3]`

`S_(15)=(15)/(2)[14+42]`

`S_(15)=(15)/(2)[56]`

`S_(15)=15* 28`

`S_(15)=420`

The sum of the first 15 term is 420.