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Geometric Sequences

Now that we know how these geometric guys work, we won't have to do the table thing anymore!

What are all the pieces?

an = 3( -2 )^( n - 1 ) ... a1 is 3 ... r is -2

Here's the general formula:

Given a geometric sequence

a1 , a2 , a3 , ... , an , ...

with a ratio of r, the nth term is

an = a1 * r^( n - 1 )

Let's try it!

What's the 23th term of this sequence?

1 / 9 , -( 1 / 3 ) , 1 , -3 , 9 , ...

Find the ratio:

r = a2 divided by a1 = -( 1 / 3 ) divided by 1 / 9 = -( 1 / 3 ) * 9 / 1 = -3

So...

an = a1 * r^( n -1 ) ... a23 = ( 1 / 9 )( -3 )^( 22 ) = -10,460,353,203 ... Yikes!
By the way, these (  ) are really important if your ratio is negative!       So, be careful!
 

TRY IT:

Find the formula for the nth term, then use it to find the 11th term of this sequence

22 / 9 , 11 / 3 , 11 / 2 , 33 / 4 , ...