A geometric series has a sum to infinity of 3, and as its second term.
Determine the two possible values of the common ratio.

Question

A geometric series has a sum to infinity of 3, and $$ \frac{2}{3} $$ as its second term. Determine the two possible values of the common ratio.

UpStudy AI · Accepted Answer

Given that the sum to infinity of a geometric series is 3, we can use the formula for the sum to infinity of a geometric series:

$$ S_{\infty} = \frac{a}{1 - r} $$

where:
- $$ S_{\infty} $$ is the sum to infinity of the geometric series,
- $$ a $$ is the first term of the geometric series,
- $$ r $$ is the common ratio of the geometric series.

We are also given that the second term of the geometric series is $$ \frac{2}{3} $$, which means the second term is $$ ar $$.

Given that $$ S_{\infty} = 3 $$ and the second term is $$ \frac{2}{3} $$, we can set up the following equations:

1. $$ \frac{a}{1 - r} = 3 $$
2. $$ ar = \frac{2}{3} $$

We need to solve these two equations to find the two possible values of the common ratio $$ r $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}\frac{a}{\left(1-r\right)}=3\ar=\frac{2}{3}\end{array}\right.$$
- step1: Remove the parentheses:
  $$\left\{ \begin{array}{l}\frac{a}{1-r}=3\ar=\frac{2}{3}\end{array}\right.$$
- step2: Solve the equation:
  $$\left\{ \begin{array}{l}a=3-3r\ar=\frac{2}{3}\end{array}\right.$$
- step3: Substitute the value of $$a:$$
  $$\left(3-3r\right)r=\frac{2}{3}$$
- step4: Simplify:
  $$r\left(3-3r\right)=\frac{2}{3}$$
- step5: Expand the expression:
  $$3r-3r^{2}=\frac{2}{3}$$
- step6: Move the expression to the left side:
  $$3r-3r^{2}-\frac{2}{3}=0$$
- step7: Factor the expression:
  $$\frac{1}{3}\left(-3r+2\right)\left(3r-1\right)=0$$
- step8: Divide the terms:
  $$\left(-3r+2\right)\left(3r-1\right)=0$$
- step9: Separate into possible cases:
  $$\begin{align}&-3r+2=0\&3r-1=0\end{align}$$
- step10: Solve the equation:
  $$\begin{align}&r=\frac{2}{3}\&r=\frac{1}{3}\end{align}$$
- step11: Calculate:
  $$r=\frac{2}{3}\cup r=\frac{1}{3}$$
- step12: Rearrange the terms:
  $$\left\{ \begin{array}{l}a=3-3r\r=\frac{2}{3}\end{array}\right.\cup \left\{ \begin{array}{l}a=3-3r\r=\frac{1}{3}\end{array}\right.$$
- step13: Calculate:
  $$\left\{ \begin{array}{l}a=1\r=\frac{2}{3}\end{array}\right.\cup \left\{ \begin{array}{l}a=2\r=\frac{1}{3}\end{array}\right.$$
- step14: Check the solution:
  $$\left\{ \begin{array}{l}a=1\r=\frac{2}{3}\end{array}\right.\cup \left\{ \begin{array}{l}a=2\r=\frac{1}{3}\end{array}\right.$$
- step15: Rewrite:
  $$\left(a,r\right) = \left(1,\frac{2}{3}\right)\cup \left(a,r\right) = \left(2,\frac{1}{3}\right)$$
The two possible values of the common ratio $$ r $$ are $$ \frac{2}{3} $$ and $$ \frac{1}{3} $$.
The two possible values of the common ratio are $$ \frac{2}{3} $$ and $$ \frac{1}{3} $$.

A geometric series has a sum to infinity of 3, and as its second term.
Determine the two possible values of the common ratio.

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A geometric series has a sum to infinity of 3, and as its second term. Determine the two possible values of the common ratio.

Upstudy AI Solution

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A geometric series has a sum to infinity of 3, and as its second term.
Determine the two possible values of the common ratio.