Question

Can someone help me?

Answer 1

see explanation

Explanation:

Given

`S_(n)` = 7n - 2n² , then

`S_(1)` = 7(1) - 2(1)² = 7 - 2 = 5

Hence a₁ = 5

Also

`S_(2)` = 7(2) = 2(2)² = 14 - 8 = 6

`S_(3)` = 7(3) - 2(3)² = 21 - 18 = 3

`S_(4)` = 7(4) - 2(4)² = 28 - 32 = - 4

Thus

a₂ = S₂ - S₁ = 6 - 5 = 1

a₃ = S₃ - S₂ = 3 - 6 = - 3

a₄ = S₄ - S₃ = - 4 - 3 = - 7

The first 4 terms of the sequence are

5, 1, - 3, - 7

There is a common difference d between consecutive terms, that is

d = 1 - 5 = - 3 - 1 = - 7 - (- 3) = - 4

The n th term of an arithmetic sequence is

`a_(n)` = a₁ + (n - 1)d

Here a₁ = 5 and d = - 4 , thus

`a_(n)` = 5 - 4(n - 1) = 5 - 4n + 4 = 9 - 4n

In conclusion

a₁ = 5

d = - 4

`t_(n)` = 9 - 4n