Question

The eighth term of an arithmetic progression is -10 and the sum of the first twenty terms is –350.

a Find the first term and the common difference.

b Given that the nth term of this progression is -97, find the value of n. ​

Answer 1

see explanation

Explanation:

The nth term of an arithmetic progression is

`a_(n)` = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

Given a₈ = - 10 , then

a₁ + 7d = - 10 → (1)

The sum of the first n terms of an arithmetic progression is

`S_(n)` =
`(n)/(2)` [ 2a₁ + (n - 1)d ]

Given

`S_(20)` = - 350 , then

`(20)/(2)` [ 2a₁ + 19d ] = - 350

10(2a₁ + 19d) = - 350

20a₁ + 190d = - 350 → (2)

Multiply (1) by - 20

- 20a₁ - 140d = 200 → (3)

add (2) and (3) term by term to eliminate a₁

0 + 50d = - 150

50d = - 150 ( divide both sides by 50 )

d = - 3

substitute d = - 3 into (1)

a₁ + 7(- 3) = - 10

a₁ - 21 = - 10 ( add 21 to both sides )

a₁ = 11

Then first term a₁ = 11 and common difference d = - 3

(b)

a₁ + (n - 1)d = - 97 , that is

11 - 3(n - 1) = - 97 ( subtract 11 from both sides )

- 3(n - 1) = - 108 ( divide both sides by - 3 )

n - 1 = 36 ( add 1 to both sides )

n = 37

That is the 37th term = - 97