In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A geometric series has common ratio $r$ and first term $a$. Given $r \neq 1$ and $a \neq 0$ (a) prove that $$S_n = \frac{a(1 - r^n)}{1 - r}$$ Given also that $S_{10}$ is four times $S_s$ (b) find the exact value of $r$.

A-Level Edexcel Maths Pure Graphs of Functions: In this question you must show a

In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 2 - 2019 - Paper 1

Question 2

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A geometric series has common rat... show full transcript

Worked Solution & Example Answer:In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 2 - 2019 - Paper 1

Step 1

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

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Answer

To prove the formula for the sum of the first $n$ terms of a geometric series, we start by writing out the series terms:

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

Next, we can express this sum in terms of the common ratio:

$S_n = a(1 + r + r^2 + ... + r^{n-1})$

Now, we multiply both sides of the equation by $(1 - r)$ :

$(1 - r)S_n = (1 - r)(a + ar + ar^2 + ... + ar^{n-1})$

This simplifies to:

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

By distributing $(1 - r)$ , we can rearrange:

$S_n - rS_n = a - ar^n$

Combining terms gives:

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

Finally, we divide both sides by $(1 - r)$ (noting that $r \neq 1$ ) to arrive at the desired result:

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

Step 2

find the exact value of $r$

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Answer

From the problem, we know that:

$S_{10} = 4S_s$

Using the formula derived in part (a), we write:

$S_{10} = \frac{a(1 - r^{10})}{1 - r}$

and

$S_s = \frac{a(1 - r^s)}{1 - r}$

Substituting these into the equation gives:

$\frac{a(1 - r^{10})}{1 - r} = 4 \frac{a(1 - r^s)}{1 - r}$

As $a \neq 0$ , we can cancel $a$ from both sides:

$\frac{1 - r^{10}}{1 - r} = 4 \frac{1 - r^s}{1 - r}$

This simplifies to:

$1 - r^{10} = 4(1 - r^s)$

Rearranging gives:

$1 - r^{10} = 4 - 4r^s$

Thus, we get:

$4r^s = 3 - r^{10}$

To find the value of $r$ , we would need the value of $s$ . Assuming $s$ is defined per specific conditions or values, we can solve for $r$ accordingly using additional equations or iterating possible values of $r$ within the range of geometric series.

In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 2 - 2019 - Paper 1

Worked Solution & Example Answer:In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 2 - 2019 - Paper 1

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

find the exact value of $r$

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