Geometric Sequences

Common ratio

9.3

SAT Prep

Quick poll!

POD preview

Give the first 5 terms of the sequence for

an = a1(3)n-1 if a1 =2

Is this formula recursive or explicit?What is the pattern in this sequence?How do we know?

POD preview

Give the first 5 terms of the sequence for

an = a1(3)n-1 if a1 =2

2, 6, 18, 54, 162

This is an explicit formula.Each term is 3 times the previous term.

Geometric sequences

If the pattern between terms in a sequence is a common ratio, then it is a geometric sequence. The ratio is r.

Recursive:

a1

an = an-1r so, r = an/an-1

What would the explicit formula be?

Geometric sequences

If the pattern between terms in a sequence is a common ratio, then it is a

geometric sequence. The ratio is r.

Explicit:

an = a1rn-1

(In other words, find the nth term by multiplying a1 by r and do that (n-1) times.)

Geometric sequences

Recursive:

a1

an = an-1r so, r = an/an-1

Explicit:

an = a1rn-1

How does these compare to the formulas for an arithmetic sequence?

Use it

Find the 10th term of our POD sequence

an = 2(3)n-1

Use it

Find the 10th term of our POD sequence

an = a1(3)n-1

a10 = 2(3)10-1 = 2(3)9 = 39366

Use it again

If the third term of a geometric sequence is 5 and the sixth term is -40, find the eighth term.

Like with arithmetic sequences, we need the first term and the change between terms.

Like we did with arithmetic sequences, we start by writing the equations. Now what?

-40 = a1(r)6-1 -40 = a1(r)5

5 = a1(r)3-1 5 = a1(r)2

Use it again

If the third term of a geometric sequence is 5 and the sixth term is -40, find the eighth term.

Once we have the equations, we can find r.

-40 = a1(r)5

5 = a1(r )2

-8 = r3 and r = -2

Use it again

If the third term of a geometric sequence is 5 and the sixth term is -40, find the eighth term.

r = -2

Once we have r, we can find a1.

5 = a1 (-2)2

5 = a1 (4)

a1 = 5/4

Use it again

If the third term of a geometric sequence is 5 and the sixth term is -40, find the eighth term.

r = -2

a1 = 5/4

Once we have r and a1, we can find the equation.

an = (5/4)(-2)n-1

And answer the question:

a8 = (5/4)(-2)8-1 = (5/4)(-2)7 = (5/4)(-128) = -160

Partial sums

Add the first 8 terms of our POD sequence

2 + 6 + 18 + 54 + 162 + 486 + 1458 + 4374

(And here’s a free vocabulary word: when we add the terms of a sequence, we call it a series. This is a finite geometric series. When we did partial sums of arithmetic sequences, those were also series.)

Partial sums

Add the first 8 terms of our sequence

2 + 6 + 18 + 54 + 162 + 486 + 1458 + 4374

= 6560

How long did that take? Want a shortcut? Not surprisingly, there are formulas.

Partial sums (finite series)

Here’s the bottom line:

Check it with our sequence:

Sn ank1

n

a1

1 rn

1 r

S8 21 38

1 3

2

6560

2

6560

Infinite sums (infinite series)

If | r | < 1, then we can determine the sum of the entire geometric series.

This is called an infinite series, and we can find the sum only in this particular case.

S a1

1 r

Infinite sums (infinite series)

An infinite series may be noted using summation notation.

If r < 0, we have something called an alternating infinite series. (Why?)

ann1

An example of an alternating series

Find the sum of the alternating geometric series

It may help to calculate the first couple of terms to verify the first term and r. Then, because | r | < 1, we can find the sum of the infinite series.

1

1

3

23

n

n

A financial example

You want to save money by setting aside a 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on.

How much would you set aside on the 15th day?

A financial example

You want to save money by setting aside a 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on.

How much would you set aside on the 15th day?

A15 = 1(2)15-1 = 214 = 16384 = $163.84

A financial example

You want to save money by setting aside a 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on.

How much have you set aside after 30 days?

A financial example

You want to save money by setting aside a 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on.

How much have you set aside after 30 days?

n = 30 S30 11 230

1 2

1073741823 $10, 737, 418.23

A fraction example

Find a rational number that corresponds to

This number can be represented as a sum.

5.4 + .027 + .00027 + .0000027 + …

5.427

A fraction example

Find a rational number that corresponds to

5.4 + .027 + .00027 + .0000027 + …

The last part looks like a geometric series where

r = .01 and a1 = .027

Since r < 1, we can find this infinite sum.

And looks like 5.4 + 3/110

= 594/110 + 3/110 = 597/110.

S .027

1 .01

.027

.99

3

110

5.427