Question

Evaluate the series 9 + 3 + 1 + . . .

Answer 1

[AI-generated answer — Bot GPT 3.5]

The given series is an arithmetic series with a common difference of −2, starting from 9.

Using the formula for the sum of an arithmetic series, we can calculate the sum Sₙ of the first n terms:

Sₙ = (n/2) * (a₁ + aₙ)
= (n/2) * (9 + aₙ)

where a₁ is the first term (9) and aₙ is the nth term.

The nth term can be determined using the formula for the nth term of an arithmetic series:

aₙ = a₁ + (n-1)d
= 9 + (n-1)(-2)
= 9 - 2n + 2
= 11 - 2n

Substituting this expression for aₙ into the formula for Sₙ:

Sₙ = (n/2) * (9 + 11 - 2n)
= (n/2) * (20 - 2n)
= 10n - n²

Therefore, the sum of the series is represented by the equation 10n - n².

However, since the series is unbounded and does not converge, we cannot determine a finite sum.

Answer 2

are you sure

Answer 3

[AI-generated answer — Bot GPT 3.5]

Apologies for the confusion in my previous response. I made a mistake in assuming the series was unbounded when it is actually bounded.

The series given is 9 + 3 + 1 + ...

We can observe that the common difference between consecutive terms is -6.

To find the sum of this arithmetic series, we can use the formula for the sum of an arithmetic series:

Sₙ = (n/2) * (a₁ + aₙ)

where Sₙ is the sum of the first n terms, a₁ is the first term, and aₙ is the nth term.

The nth term can be determined using the formula for the nth term of an arithmetic series:

aₙ = a₁ + (n-1)d
= 9 + (n-1)(-6)
= 9 - 6n + 6
= 15 - 6n

Substituting this expression for aₙ into the formula for Sₙ:

Sₙ = (n/2) * (9 + 15 - 6n)
= (n/2) * (24 - 6n)
= 12n - 3n²

Therefore, the sum of the series 9 + 3 + 1 + ... can be represented by the equation 12n - 3n².