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 seq

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​

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Bình An 3 years 2021-08-25T01:53:57+00:00 1 Answers 2 views 0

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    2021-08-25T01:55:40+00:00

    Answer:

    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.

    Step-by-step 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 nth 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 = \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 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 = \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, (1 + (3 - 1) \times 2) = 5, and (1 + (13 - 1) \times 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 \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}.

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