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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 5: sum of infinite convergent geometric sequences. 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 this infinite convergent geometric sequence
$$64, 32, 16, 8, 4 \ldots$$

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2

Evaluate this infinite convergent geometric series
$$81+27+9+3+ \ldots$$

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3

A geometric sequence has eight as its first term and a common ratio of three-quarters. Find the sum of the terms of the sequence as the number of terms tends to infinity.

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4

Only one of the following geometric series converges. What does it converge to?
$$24 + 30 + 37.5 + 46.875 + \ldots\\24 + 18 + 13.5 + 10.125 + \ldots\\24 + 27 + 30.375 + 34.171875 + \ldots$$

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5

Only one of the following geometric series converges. What does it converge to?
$$48 -60 + 75 -93.75 + \ldots\\48 -84 + 147 -257.25 + \ldots\\48 -42 + 36.75 -32.15625 + \ldots$$

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6

The third term of a geometric sequence is 64. The sixth term is 8. Find the sum to infinity of the sequence.

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7

The sum to infinity of a geometric sequence is 5. Its first term is five times its common ratio. Find the second term of the sequence.

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8

The sum to infinity of a geometric sequence is 81 and the sum of the first four terms is 65. Find the third term of the sequence if it is known to be greater than 50.

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9

The following geometric series converges. What is the smallest number that \(x\) must be less than?
$$ (3-x) + (3-x)^2 + (3-x)^3 + \ldots$$

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10

The second term of a geometric sequence is \(-\frac{4}{5} \). The sum to infinity of the sequence is \( 7\frac{3}{11} \). What is the first term of the sequence?

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Check

This is Geometric Sequences level 5. You can also try:
Arithmetic Sequences Level 1 Level 2 Level 3 Level 4 Quadratic Sequences

Instructions

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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 - Find the first five terms of the sequence given the formula

Level 4 - Mixed questions about geometric sequences and their sums

Level 5 - 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.

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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 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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