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​

Respuesta :

Answer:

The first three terms of the geometry sequence would be [tex]1[/tex], [tex]5[/tex], and [tex]25[/tex].

The sum of the first seven terms of the geometric sequence would be [tex]127[/tex].

Step-by-step explanation:

1.

Let [tex]d[/tex] denote the common difference of the arithmetic sequence.

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

The question states that the first term of this arithmetic sequence is [tex]a_1 = 1[/tex]. Hence:

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

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

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

[tex]\displaystyle r = \frac{1 + 2\, d}{1} = 1 + 2\, d[/tex].

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

[tex]\displaystyle r = \frac{1 + 12\, d}{1 + 2\, d}[/tex].

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

[tex]\displaystyle 1 + 2\, d = \frac{1 + 12\, d}{1 + 2\, d}[/tex].

[tex](1 + 2\, d)^{2} = 1 + 12\, d[/tex].

[tex]d = 2[/tex].

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

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

2.

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

[tex]\displaystyle \frac{a_1 \, \left(1 - r^{n}\right)}{1 - r}[/tex].

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

Hence, the sum of the first [tex]n = 7[/tex] terms of this geometric sequence would be:

[tex]\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}[/tex].