Question

The first, third and thirteenth terms of an arithmetic sequence are the first 3 terms of a geometric sequence. If the first term of both sequences is 1, determine:

1.) the first three terms of the geometric sequence if r > 1

2.) the sum of 7 terms of the geometric sequence if the sequence is 1, 5, 25​

Answer 1

The first three terms of the geometry sequence would be
`1`,
`5`, and
`25`.

The sum of the first seven terms of the geometric sequence would be
`127`.

Explanation:

1.

Let
`d` denote the common difference of the arithmetic sequence.

Let
`a_1` denote the first term of the arithmetic sequence. The expression for the
`n`th term of this sequence (where
`n\!` is a positive whole number) would be
`(a_1 + (n - 1)\, d)`.

The question states that the first term of this arithmetic sequence is
`a_1 = 1`. Hence:

• The third term of this arithmetic sequence would be
  `a_1 + (3 - 1)\, d = 1 + 2\, d`.
• The thirteenth term of would be
  `a_1 + (13 - 1)\, d = 1 + 12\, d`.

The common ratio of a geometric sequence is ratio between consecutive terms of that sequence. Let
`r` denote the ratio of the geometric sequence in this question.

Ratio between the second term and the first term of the geometric sequence:

`\displaystyle r = (1 + 2\, d)/(1) = 1 + 2\, d`.

Ratio between the third term and the second term of the geometric sequence:

`\displaystyle r = (1 + 12\, d)/(1 + 2\, d)`.

Both
`(1 + 2\, d)` and
`\left(\displaystyle (1 + 12\, d)/(1 + 2\, d)\right)` are expressions for
`r`, the common ratio of this geometric sequence. Hence, equate these two expressions and solve for
`d`, the common difference of this arithmetic sequence.

`\displaystyle 1 + 2\, d = (1 + 12\, d)/(1 + 2\, d)`.

`(1 + 2\, d)^(2) = 1 + 12\, d`.

`d = 2`.

Hence, the first term, the third term, and the thirteenth term of the arithmetic sequence would be
`1`,
`(1 + (3 - 1) * 2) = 5`, and
`(1 + (13 - 1) * 2) = 25`, respectively.

These three terms (
`1`,
`5`, and
`25`, respectively) would correspond to the first three terms of the geometric sequence. Hence, the common ratio of this geometric sequence would be
`r = 25 /5 = 5`.

2.

Let
`a_1` and
`r` denote the first term and the common ratio of a geometric sequence. The sum of the first
`n` terms would be:

`\displaystyle (a_1 \, \left(1 - r^(n)\right))/(1 - r)`.

For the geometric sequence in this question,
`a_1 = 1` and
`r = 25 / 5 = 5`.

Hence, the sum of the first
`n = 7` terms of this geometric sequence would be:

`\begin{aligned} & (a_1 \, \left(1 - r^(n)\right))/(1 - r)\\ &= (1 * \left(1 - 2^(7)\right))/(1 - 2) \\ &= ((1 - 128))/((-1)) = 127 \end{aligned}`.