Moderate -0.3 This is a standard C2 sequences and series question requiring routine application of arithmetic and geometric progression formulas. Part (a) is a bookwork proof that should be memorized, parts (b-c) involve straightforward substitution into the arithmetic series formula, and part (d) uses the basic geometric sequence formula. While multi-part with 13 total marks, each component is procedural with no novel problem-solving required, making it slightly easier than average.
An arithmetic series has first term a and common difference d. Prove that the sum of the first n terms of the series is
$$\frac{1}{2}n[2a + (n - 1)d].$$ [4 marks]
A company made a profit of £54000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £d. This model predicts total profits of £619200 for the 9 years 2001 to 2009 inclusive.
Find the value of d. [4 marks]
Using your value of d,
find the predicted profit for the year 2011. [2 marks]
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 £54000,
find the predicted profit for the year 2011. [3 marks]
(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
$$\frac{1}{2}n[2a + (n - 1)d].$$ [4 marks]
A company made a profit of £54000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £d. This model predicts total profits of £619200 for the 9 years 2001 to 2009 inclusive.
(b) Find the value of d. [4 marks]
Using your value of d,
(c) find the predicted profit for the year 2011. [2 marks]
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 £54000,
(d) find the predicted profit for the year 2011. [3 marks]
\hfill \mbox{\textit{Edexcel C2 Q3 [13]}}