Geometric Sequences Examples

A geometric sequence has a₁ = 1 and r = 2. Find the first four terms of the sequence.

To get the next term, we multiply the term we're on by r = 2.

This means

a₂ = a₁(2) = 2

a₃ = a₂(2) = 4

a₄ = a₃(2) = 8.

This isn't a very useful form for seeing patterns. However, if we write each term as its prime factorization, then we can see what's going on.

From this table, we can conclude that the n^th term is

a_n = 2^{n – 1}.

Multiplying numbers together hides patterns. Since we like patterns so much, we are going to keep expressions in factored form. It'll be easier to see what's going on.

A geometric sequence has a₁ = 5 and r = 2. Find

(a) a₂

(b) a₃

(c) a₄

(d) a_n

To get from one term to the next, we multiply by r = 2. We're NOT going to multiply the numbers all together. We're going to keep the terms in factored form so we can see what's going on.

(a) a₂ = a₁(2) = 5 × 2

(b) a₃ = a₂ × 2
= 5 × 2 × 2
= 5 × 2²

(c) a₄ = a₃ × 2
= 5 × 2 × 2 × 2
= 5 × 2³

(d) Look at the terms we have so far:

We must have a_n = 5 × 2^{n – 1}.

A geometric sequence has a = 2 and r = 10. Find a₅.

We plug the numbers we were given into the nice formula:

a_n = ar^{n – 1}

a₅ = (2)(10)⁴

= 20,000.

That's all there is to it.

A geometric sequence has a₉ = 40 and a₁₀ = 80. Find a and r.

We know that r is the ratio between successive terms, so

Then we can use the formula to find a. We know a₉ = 40 and r = 2, so

a_n = ar^{n – 1}

a₉ = a(2)⁸

40 = a(256)

Find a and r for the geometric sequence with a₈ = -0.78125 and a₉ = 0.390625.

The common ratio is

Plugging this and a₈ into the formula, we get

a_n = ar^{n – 1}

a₈ = a(-0.5)⁷

-0.78125 = a(-0.0078125)

We conclude that

Let's check this answer by finding a₈ and a₉ now that we think a = 100 and r = -0.5.

a_n = ar^{n – 1}

a₈ = (100)(-0.5)⁷

= -0.78125

a₉ = (100)(-0.5)⁸

= 0.390625.

These are the values we were initially given for a₈ and a₉, which means we correctly found

a = 100

r = -0.5

Find a and r for the geometric sequence with a₅ = 81 and a₉ = 410.0625.

Find the ratio between the terms. Since the terms aren't consecutive, this value is not r:

It takes

9 – 5 = 4

multiplications to get from a₅ to a₉.

This means

a₉ = a₅r⁴

and so

This means the number we found when we divided a₉ by a₅ is r⁴. If we take its 4th root, we get r.

r = (5.0625)^1/4 = 1.5.

Now we proceed as usual to find a.

a_n = ar^{n – 1}

a₅ = a(1.5)⁴

81 = a(5.0625)

Then we get

Find a and r for the geometric sequence with a₅ = 6.561 and a₈ = 4.782969.

First we find the ratio of the terms:

Since it takes three multiplications to get from a₅ to a₈, we take the third root to get r:

r = 0.729^1/3 = 0.9.

Finally, we use the formula to find a.

a_n = ar^{n – 1}

a₅ = a(0.9)⁴

6.561 = a(0.9)⁴

We get

Be careful. When doing these problems, don't round. Use all the digits at each step, or you'll probably get the wrong answer.