Answer:
see explanation
Step-by-step explanation:
The nth term of an arithmetic progression is
[tex]a_{n}[/tex] = 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
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] [ 2a₁ + (n - 1)d ]
Given
[tex]S_{20}[/tex] = - 350 , then
[tex]\frac{20}{2}[/tex] [ 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
