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Arithmetic and Geometric Sequences

g(n) = g*(r^(n-1))

Now we can do an example to see how this all pieces together.

Example 3: Consider the following sequence.

10, 30, 90, 270, ...

Find g(1). What is the common ratio? Find the general formula for this sequence. What is g(8)?

First, try this yourself, then look at the solution when you're finished.

Solution:

g(1) = 10

This comes straight from 10 being the first term in the sequence.

We can get the common ratio by dividing two consecutive terms in the sequence.

r = 270 / 90

r = 3

Putting this information together, we get the general formula for the sequence.

g(n) = g*(r^(n-1))

g(n) = 10*(3^(n-1))

Finally, the 8th term of this sequence is 

g(8) = 10*(3^(8-1))

g(8) = 10*(3^(7))

g(8) = 10*2187

g(8) = 21870

This concludes everything we can do with both geometric and arithmetic sequences. A key thing to note about arithmetic sequences is that the common difference never changes. So, if you are looking at a sequence that you think is an arithmetic sequence. Check the common difference using different consecutive terms of the sequence. If you notice that the common difference is changing for different successive terms, then you are most likely working with a geometric sequence. Now, before I go, I have one last example for us to solve.

Example 4: Suppose we are given three terms of a sequence. 

f(4) = 9/2

f(5) = 3/4

f(6) = 1/8

Find f(1). Find the general formula for the sequence.

Try this problem out yourself first. When you are finished, then look at the solution.

Solution:

So, we know that we have a sequence, but we don't know what kind of sequence it is. So first, let's check for a common difference between two successive terms.

d = (3/4) - (9/2) = -15/4

Hmmm. Seems interesting. Let's try another pair.

d = (1/8) - (3/4) = -5/8

Well, the common difference is changing, so we must be working with a geometric sequence. But to be sure, let's just double-check.

r = (1/8) / (3/4) = (1/8) * (4/3)

r = (1/2) * (1/3) = 1/6

Okay, let's check another pair.

r = (3/4) / (9/2) = (3/4) * (2/9)

r = (1/2) * (1/3) = 1/6

So our r stayed the same. So now we know for certain that this is a geometric sequence.

To find g(1) let's remember the formula for the nth term of a geometric sequence.

g(n) = g*(r^(n-1))

Well, we have 3 different terms to choose from, so let's choose f(5).

f(5) = g*(r^(5-1))

3/4 = g* ((1/6)^4)

3/4 = g/(6^4)

Now we solve for g (=g(1)).

g = (3/4)*(6^4)

g = 972

g(1) = 972

Now we have everything we need to create the general formula for this sequence.

f(n) = 972*((1/6)^(n-1))

Simplifying a small bit, we get our final answer.

f(n) = 972/(6^(n-1))

Be sure to remember the general formulas for arithmetic and geometric sequences. 

a(n) = a(1) + (n-1)*d

g(n) = g(1) * r^(n-1)

Sources:

Stapel, E. (n.d.). Arithmetic & GEOMETRIC SEQUENCES. Retrieved February 06, 2021, from https://www.purplemath.com/modules/series3.htm