Question

Perform the following to demonstrate you can work with geometric sequences:a. Write 5 numbers that are an example of a geometric sequence.b. Show the value of the common ratio.c. Write an equation that would give you the value of the 23rd term in the sequenced. Find the sum of the first 10 terms of the sequence. Show your work or explain you answer.

Answer 1

a. 3, 9, 27, 81, 243

b. 3

c. 3^n

d. 88572

Explanation:

a.

A geometric sequence has the following form:

`r,r^2,r^3,...,r^n`

We calculate the first 5 numbers like this:

`\begin{gathered} 3,3^2,3^3,3^4,3^5 \\ \\ 3,9,27,81,243 \end{gathered}`

b.

The common ratio is calculated like this:

`\begin{gathered} (243)/(81)=3 \\ \\ (81)/(27)=3 \\ \\ (27)/(9)=3 \\ \\ (9)/(3)=3 \\ \\ \text{ Therefore, the common ratio is 3} \end{gathered}`

c.

We have that the equation of the sequence is the following:

`\begin{gathered} a_n=3^n \\ \\ \text{ when n = 23, we replacing:} \\ \\ a_(23)=3^(23) \\ \\ \text{ the equation that would give you the value of the 23rd term in the sequence is }3^n \end{gathered}`

d.

The sum of the first 10 terms is given as follows:

`\begin{gathered} s=3+3^2+3^3+...+3^(10) \\ \\ \text{ The sum can be established using the following formula:} \\ \\ s=(r(r^n-1))/(r-1) \\ \\ \text{ We replacing:} \\ \\ s=(3\cdot(3^(10)-1))/(3-1)=(3(59049-1))/(2)=(3\cdot\:59048)/(2) \\ \\ s=88572 \\ \\ \text{ The sum of first 10 terms is 88572} \end{gathered}`