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​

The first three terms of the geometry sequence would be , , and .

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

Step-by-step explanation:

Let denote the common difference of the arithmetic sequence.

Let denote the first term of the arithmetic sequence. The expression for the th term of this sequence (where is a positive whole number) would be .

The question states that the first term of this arithmetic sequence is . Hence:

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

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

.

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

.

Both and are expressions for , the common ratio of this geometric sequence. Hence, equate these two expressions and solve for , the common difference of this arithmetic sequence.

.

.

.

Hence, the first term, the third term, and the thirteenth term of the arithmetic sequence would be , , and , respectively.

These three terms (, , and , respectively) would correspond to the first three terms of the geometric sequence. Hence, the common ratio of this geometric sequence would be .

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

.

For the geometric sequence in this question, and .

Hence, the sum of the first terms of this geometric sequence would be:

.