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

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 $n$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:

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

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:

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