8. (a) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]\). A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
(b) Find the value of \(d\). Using your value of \(d\),
(c) find the predicted profit for the year 2011. An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06 . Using this alternative model and again taking the profit in 2001 to be \(\pounds 54000\),
(d) find the predicted profit for the year 2011.

## Question 8:

### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $S = a + (a+d) + (a+2d) + \ldots + [a+(n-1)d]$ | B1 | |
| $S = [a+(n-1)d] + [a+(n-2)d] + \ldots + a$ | M1 | |
| Add: $2S = n[2a+(n-1)d] \Rightarrow S = \frac{1}{2}n[2a+(n-1)d]$ | M1 A1 | (4) |

### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 54000$ and $n = 9$ | B1 | |
| $619200 = \frac{1}{2}\times 9 \times (2\times 54000 + 8d)$ | M1 A1ft | |
| $d = 3700$ | A1 | (4) |

### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a + (n-1)d = a + 10d = 54000 + 10d = £91000$ | M1 A1 | (2) |

### Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $ar^{n-1} = 54000 \times 1.06^{10}$ (ft their $n$) | M1 A1ft | |
| $= £96700$ (or £97000) | A1 | (3) |

8. (a) An arithmetic series has first term $a$ and common difference $d$. Prove that the sum of the first $n$ terms of the series is $\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$.

A company made a profit of $\pounds 54000$ in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference $\pounds d$. This model predicts total profits of $\pounds 619200$ for the 9 years 2001 to 2009 inclusive.\\
(b) Find the value of $d$.

Using your value of $d$,\\
(c) find the predicted profit for the year 2011.

An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06 . Using this alternative model and again taking the profit in 2001 to be $\pounds 54000$,\\
(d) find the predicted profit for the year 2011.

\hfill \mbox{\textit{Edexcel C2  Q8 [13]}}