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

Answer:

a) The first term is a=7

b) The common difference is d=3

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

Step-by-step 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, $S_n=\frac{n}{2}[2a+(n-1)d]$

$S_{15}=\frac{15}{2}[2(7)+(15-1)3]$

$S_{15}=\frac{15}{2}[14+(14)3]$

$S_{15}=\frac{15}{2}[14+42]$

$S_{15}=\frac{15}{2}[56]$

$S_{15}=15\times 28$

$S_{15}=420$

The sum of the first 15 term is 420.