A geometric series is a + ar + ar² + ... (a) Prove that the sum of the first n terms of this series is given by $$S_n = \frac{a(1 - r^n)}{1 - r}$$ The third and fifth terms of a geometric series are 5.4 and 1.944 respectively and all the terms in the series are positive. For this series find, (b) the common ratio, (c) the first term, (d) the sum to infinity.

A-Level Edexcel Maths Pure Equation of a Straight Line: A geometric series is a + ar + a

A geometric series is a + ar + ar² + .. - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 3

Question 3

A geometric series is a + ar + ar² + ... (a) Prove that the sum of the first n terms of this series is given by $$S_n = \frac{a(1 - r^n)}{1 - r}$$ The third and f... show full transcript

Worked Solution & Example Answer:A geometric series is a + ar + ar² + .. - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 3

Step 1

Prove that the sum of the first n terms of this series is given by

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Answer

To prove the formula for the sum of the first n terms of a geometric series, we start with the definition of the sum:

$S_n = a + ar + ar^2 + \ldots + ar^{n-1}$

Multiplying both sides by the common ratio $r$ , we get:

$rS_n = ar + ar^2 + ar^3 + \ldots + ar^n$

Subtracting these two equations:

$S_n - rS_n = a - ar^n$

Factoring out $S_n$ :

$S_n(1 - r) = a(1 - r^n)$

Rearranging gives:

$S_n = \frac{a(1 - r^n)}{1 - r}$

Step 2

the common ratio

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Answer

Let the common ratio be $r$ . The third term is given by:

$ar^2 = 5.4$

And the fifth term is:

$ar^4 = 1.944$

Dividing the second equation by the first:

$\frac{ar^4}{ar^2} = \frac{1.944}{5.4}$

This simplifies to:

$r^2 = \frac{1.944}{5.4} = 0.36$

Taking the square root gives:

$r = 0.6$ .

Step 3

the first term

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Answer

Using the common ratio found, $r = 0.6$ . Substituting $r$ back into the equation from part (b):

$ar^2 = 5.4$

Thus, we have:

$a(0.6)^2 = 5.4$

Calculating gives:

$0.36a = 5.4$

Hence,

$a = \frac{5.4}{0.36} = 15.$

Step 4

the sum to infinity

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Answer

The sum to infinity of a geometric series is given by the formula:

$S = \frac{a}{1 - r}$

Substituting the values of $a$ and $r$ :

$S = \frac{15}{1 - 0.6} = \frac{15}{0.4} = 37.5.$

A geometric series is a + ar + ar² + .. - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 3

Worked Solution & Example Answer:A geometric series is a + ar + ar² + .. - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 3

Prove that the sum of the first n terms of this series is given by

the common ratio

the first term

the sum to infinity

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