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Geometric Sequences

An exercise on geometric sequences including finding the nth term and the sum of n terms.
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This is level 3: mixed questions about geometric sequences and their sums. You will be awarded a trophy if you get at least 7 answers correct and you do this activity online.

1

Find the sum of the first six terms of the geometric series that begins:
$$2, 4, 8, 16\ldots$$

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2

Find the sum of the first ten terms of the geometric series that begins:
$$7, 21, 63, 189\ldots$$

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3

A geometric sequence has 5 as its first term. The common ratio is two. Calculate the sum of the first eight terms of this sequence .

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4

A geometric sequence has 6 as its first term. The common ratio is one third. Calculate the sum of the first ten terms of this sequence giving your answer correct to three significant figures.

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5

A geometric sequence has first term 3 and fourth term 24. Calculate the sum of the first five terms of this sequence.

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6

A geometric sequence has first term 20 and fourth term 2.5. Calculate the sum of the first five terms of this sequence.

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7

Find \(X\) if
$$X = 2 + 4 + 8 + 16 + \ldots + 8192$$

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8

Find \(Y\) if
$$Y = - 32 + 16 - 8 + 4 - 2 \ldots -\frac{1}{2}$$

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9

A geometric sequence has first term 16 and common ratio \(\frac12\). The sum of the first n terms of the sequence is 31. Find n.

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10

A geometric sequence has first term 4 and common ratio 2. The sum of the first n terms of the sequence is 65532. Find n.

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Check

This is Geometric Sequences level 3. You can also try:
Arithmetic Sequences Level 1 Level 2 Level 4 Quadratic Sequences

Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.
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Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

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"This is a great memory aid which could be used for formulae or key facts etc - in any subject area. The PICTURE is such an aid to remembering where each number or group of numbers is - my pupils love it!
Thanks"

Comment recorded on the 28 May 'Starter of the Day' page by L Smith, Colwyn Bay:

"An absolutely brilliant resource. Only recently been discovered but is used daily with all my classes. It is particularly useful when things can be saved for further use. Thank you!"

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Description of Levels

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Level 1 - Find the next term of these geometric sequences

Level 2 - Find a given term of these geometric sequences

Level 3 - Mixed questions about geometric sequences and their sums

Level 4 - Sum of infinite convergent geometric sequences

Missing Terms - Find the missing terms of arithmetic, geometric and Fibonacci-type sequences in this self marking quiz.

Exam Style questions are in the style of GCSE or IB/A-level exam paper questions and worked solutions are available for Transum subscribers.

Arithmetic Sequences - A similar exercise on arithmetic sequences.

Sigma - Practise using the sigma notation to find the sum of various number series.

More on this topic including lesson Starters, visual aids and investigations.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Geometric Sequences

Here is a reminder of some facts that may help you answering the questions in this exercise.

An geometric sequence, sometimes called a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 5, 10, 20, 40, ... is a geometric sequence with common ratio 2.

The first term of the sequence can be written as u1

The nth term of the sequence can be written as un

The common ratio is usually written as r

The formula for finding the nth term is un=u1r(n-1)

The excellent video above is from Corbettmaths.

The formula for finding the sum of \(n\) terms is:

$$ S_n = \dfrac{u_1(r^n-1)}{r-1}$$

The sum of an infinite geometric sequence is:

$$ S_\infty = \dfrac{u_1}{1-r}, \quad |r| \lt 1 $$

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