Week 2 – Geometric Sequences

Posted on by shannonf2015

This past week, I have learned many different formulas and techniques to solve geometric sequences such as finding the sum, a term, and common difference. One of the most important concepts I have learned was how to find a term when given two terms that are not consecutive.

My geometric sequence problem:

In a geometric sequence, the third term is 189 and the sixth term is 5103. Find the eighth term.

Terms:

r = common ratio

a = first term

Step 1: Write down the given values.

t_3 = 189

t_6 = 5103

t_8 = ?

Step 2: Decide which equation would work best with the information given and what info we are missing.

= t_n={ar^{n-1}}

As we can see in this equation, we are missing a=t_1 and r=common ratio.

To find these values, we must first start by finding r to be able to find the first term.

Since we are given the values for two terms: t_3 and t_6, we can find the relation between them in their geometric sequence to find r.

In this image, it shows that to start from the third term, it takes three multiples of the common ratio to get to the sixth term. The equation to find r would look like this:

= t_3\times{r}^3=t_6

Now plug in the values given: t_3 = 189, t_6 = 5103

= {189}\times{r}^3=5103

To find r, we must isolate {r}^3. To isolate {r}^3, you must divide 189 on both sides of the equation.

\frac{189}{189}\times{r}^3={5103}\div{189}

{r}^3 = 27

\sqrt[3]{r} = \sqrt[3]{27}

r = 3

Step 3: Now that we have the value of r, we can find t_1.

Starting from t_3 to go to t_1, we must divide by how many times they are spaced out by the common ratio, as it’s going backwards.

Equation would look like this: t_3\div{r}^2 = t_1

189\div{3}^2 = t_1

189\div{9} = t_1

21 = t_1

Now that we have found t_1 and r, we can plug in all the values in the equation.

*We must remember to use Bedmas when solving the equation.*

Back to the equation: t_n={ar^{n-1}}

a = 21

r = 3

n = 8

= t_n = t_8

= t_8={(21)(3)^{8-1}}

= t_8={(21)(3)^{7}}

= t_8={(21)(2187)}

= t_8={45 927}

The eighth term of the geometric sequence is 45 927.

 

 

 

 

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