Geometric Sequences

  These are sequences where you go from term to term by multiplying by a common ratio. The sequence in the image 1, 2 , 4 , 8 has a common ratio of 2 since we multiply the previous term by 2. There are formula in the booklet to help you with this topic, but it is better not to be too reliant on them. This topic is linked to exponential growth and compound interest. Exam questions often use other areas of the mathematics course. We will get practice in all of these ideas on this page


Key Concepts

On this page, you should learn to

  • Use the nth term of a geometric sequence
  • Use the nth term of a geometric series
  • Use \(\sum\) notation for sums
  • Understand applications such as compound interest and depreciation
  • Find the sum of an infinite convergent series

Essentials

The following videos will help you understand all the concepts from this page

Understanding Geometric Sequences

Due to the style of the IB questions, it is not good to become reliant on the formula in the booklet. In the following video, we will gain a conceptual understanding of where the formula for the nth term of geometric sequences come from.

Notes from the video

In the following applet, you can explore the effect changing r (and U1) has on the sequence.

What happens when

  1. r > 1
  2. r = 1
  3. 0 < r < 1
  4. r = 0
  5. -1 < r < 0
  6. r = -1
  7. r < -1

Finding Missing Terms

Being reliant on the formula for the nth term of a sequence can sometimes make the problems more difficult to solve. In the following video, it is far easier just to think about the problem conceptually.

A geometric sequence has second term 36 and fourth term 16.

Find the first term.

Notes from the video

How many Terms?

In the following video, we look at an example in which we have to find the number of terms in a geometric sequence. This requires us to solve an equation with indices. There are several ways of doing this: using logarithms, the table function on our calculator or a graphical method. The example is about compound interest.

$10 000 is invested in a bank receiving 4% interest at the end of each year.

How long does it take before the investment doubles in value?

Notes from the video

Algebraic Problem

In the following video, we need to use our algebraic skills to find an unknown

\(2k+5 \ ,\ k+5 \ ,\ k-1\) are consecutive terms of a geometric sequence. Find k

Notes from the video

Sum of Geometric Series

When we add terms of a sequence together, we call it a series

geometric sequence 1 , 2 , 4 , 8

geometric series 1 + 2 + 4 + 8

There are two formulae for the sum to n terms of a geometric series

\({ S }_{ n }=\frac { { U }_{ 1 }({ r }^{ n }-1) }{ r-1 } \) ... useful when r > 1

\({ S }_{ n }=\frac { { U }_{ 1 }({ 1-r }^{ n }) }{ 1-r } \) ... useful when r < 1

Sum to Infinity

If -1 < r < 1 , then the series is convergent

Here, r = 0.5 and the series converges.


When the series converges, we can find the sum to infinity

\({ S }_{ \infty }=\frac { { U }_{ 1 } }{ r-1 } \)


In the following video, we look at a typical exam-style question which involves the sum of a geometric series. In particular, we look at a very useful technique for solving simultaneous equations:

The sum of the first three terms of a geometric series is 61.

The sum to infinity is 125.

Find the common ratio.

Notes from the video

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

 Here is a quiz about the financial applications of geometric sequences


START QUIZ!

Exam-style Questions

Question 1

The nth term of a geometric sequence is Un , where Un =\(48(\frac{1}{4})^n\)

a) Find U1

b) Find the sum to infinity of the series

Hint

Full Solution

 

Question 2

The first term of a geometric series is 10. The sum to infinity is 50.

a) Find the common ratio

The nth term is Un

b) Find the value of \(\sum _{ n=1 }^{ 20 }{ { U }_{ n } } \)

Hint

Full Solution

 

Question 3

Three terms of a geometric sequence are \(x+6 \ , \ 12 \ ,\ x-1\)

Find the possible values of x

Hint

Full Solution

 

Question 4

Consider the geometric sequence where the first term is 45 and the second term is 36.

a) Find the least value of n such that the nth term of the sequence is less than 1

b) Find the least value of n such that the sum of the first n terms of the sequence is more than 200.

c) Find the sum to infinity.

Hint

Full Solution

 

Question 5

The sum to infinity of a geometric series is 27.

The sum of the first 3 terms is 19.

Find the common ratio

Hint

Full Solution

 

Question 6

The 2nd, 3rd and 6th terms of an arithmetic sequence with common difference \(d, \ d\neq 0\) form the first three terms of a geometric sequence, with common ratio, r.

The 1st term of the arithmetic sequence is -2.

a) Find d.

The sum of the first n terms of the geometric sequence exceeds the sum of the first n terms of the arithmetic sequence by at least 1000.

b) Find the least value of n for which this occurs.

Hint

Full Solution

 

Question 7

U1 = cosx ,U2 = sin2x are the first two terms of a geometric sequence, \(-\frac{\pi}{2}<x<\frac{\pi}{2}\)

a) Find U3 in terms of cosx

 

 

b) Find the set of values of x for which the geometric series converges

Hint

Full Solution

 

 

Question 8

a) Jessica takes out a loan of $200 000 to buy an appartment. The interest rate is 4% and is calculated at the end of each year. Calculate to the nearest dollar the amount Jessica would owe the bank after 15 years.

b) In order to pay of the loan, she pays $P into a bank at the end of each year. She receives an interest rate of 2.5% per year. Find the amount saved after 15 years.

c) What must be the value of P so that she has saved enough money to pay off the loan.

Hint

Full Solution

 

MY PROGRESS

How much of Geometric Sequences have you understood?