6. (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.

## Question 6:

### Part (a)
$S = a + (a+d) + (a+2d) + \ldots + [a+(n-1)d]$ | 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 marks)

### Part (b)
$a = 54000$ and $n = 9$ | B1 |

$619200 = \frac{1}{2} \times 9 \times (2 \times 54000 + 8d)$ | M1 A1ft |

$d = 3700$ | A1 | (4 marks)

### Part (c)
$a + (n-1)d = a + 10d = 54000 + 10d = £91000$ | M1 A1 | (2 marks) **(8 marks)**

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6. (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.\\

\hfill \mbox{\textit{Edexcel C1  Q6 [10]}}