Mathematics, 17.03.2021 23:50 aaliyahlopez85

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

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Answer from: MINDYDELANO

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

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

Step-by-step explanation:

Let denote the common difference of the arithmetic sequence.

Let a_1 denote the first term of the arithmetic sequence. The expression for the 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 denote the ratio of the geometric sequence in this question.

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

$\displaystyle r = \frac{1 + 2\, d}{1} = 1 + 2\, d$ .

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

$\displaystyle r = \frac{1 + 12\, d}{1 + 2\, d}$ .

Both $(1 + 2\, d)$ and $\left(\displaystyle \frac{1 + 12\, d}{1 + 2\, d}\right)$ 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.

$\displaystyle 1 + 2\, d = \frac{1 + 12\, d}{1 + 2\, d}$ .

$(1 + 2\, d)^{2} = 1 + 12\, d$ .

d = 2 .

Hence, the first term, the third term, and the thirteenth term of the arithmetic sequence would be , $(1 + (3 - 1) \times 2) = 5$ , and $(1 + (13 - 1) \times 2) = 25$ , 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 r = 25 /5 = 5 .

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

$\displaystyle \frac{a_1 \, \left(1 - r^{n}\right)}{1 - r}$ .

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:

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

Answer from: Quest

1) c 2143.6 cm

2) c 113.0 m

3) a 33.5 ft

4) d 2 cm

5) c 20 mm

6) 180 in

7) d 120 ft

Answer from: Quest

The answer would he -4,-2 because the points are just on the opposite side. since the x axis is +4 it's going to he -4 and since the y axis is -2 it stays there because whaen rotated it stays in the third quadrant
Hello! could someone with this problem?

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The first, third and thirteenth terms of an arithmetic sequence are the first 3 terms of a geometric...