Question

Find the sum of first seventeen numbers of an ap whose 4th and 9th tearm are -15 and -30

Answer 1

The sum of first seventeen terms is - 510

Explanation:

Given as :

The 4th term of an A.P =
`t_4` = - 15

The 9th term of an A.P =
`t_9` = - 30

For an arithmetic progression

The nth term is given as
`t_n` = a + ( n - 1)×d

Where a is the first term and d is the common difference between numbers

So, For 4th term

`t_n` = a + ( n - 1)×d

Or,
`t_4` = a + ( n - 1)×d

- 15 = a + ( 4 - 1)×d

Or, - 15 = a + 3 d .........1

So, For 9th term

`t_n` = a + ( n - 1)×d

`t_9` = a + ( n - 1)×d

- 30 = a + ( 9 - 1)×d

Or, - 30 = a + 8 d .........2

Solve eq 1 and 2

( a + 8 d ) - ( a + 3 d ) = - 30 - ( - 15)

or, ( a - a ) + ( 8 d - 3 d ) = - 30 + 15

or, 0 + 5 d = - 15

∴ d = -
`(15)/(5)` = - 3

Now, put the value of d in eq 1

I.e - 15 = a + 3 × ( - 3)

Or. - 15 = a - 9

∴ a = -15 + 9 = - 6

Now The sum of nth term is written as :

`s_n` =
`(n)/(2)` × [ 2 × a + ( n - 1 )×d ]

Where n is the nth term

a is the first term

d is the common difference

So For n = 17th term

`s_17` =
`(17)/(2)` × [ 2 × ( - 6) + ( 17 - 1 )×( - 3) ]

Or,
`s_17` =
`(17)/(2)` × [ - 12 - 48 ]

Or,
`s_17` =
`(17)/(2)` × ( - 60 )

Or,
`s_17` = 17 × ( - 30)

∴
`s_17` = - 510

Hence The sum of first seventeen terms is - 510 Answer