In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers than the year before, so that the shop sold 160 computers in 2001, 170 computers in 2002, and so on forming an arithmetic sequence. (a) Show that the shop sold 220 computers in 2007. (b) Calculate the total number of computers the shop sold from 2000 to 2013 inclusive. In the year 2000, the selling price of each computer was £900. The selling price fell by £20 each year, so that in 2001 the selling price was £880, in 2002 the selling price was £860, and so on forming an arithmetic sequence. (c) In a particular year, the selling price of each computer in £s was equal to three times the number of computers the shop sold in that year. By forming and solving an equation, find the year in which this occurred.

Photo AI

In the year 2000 a shop sold 150 computers - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 1

Question 10

In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers than the year before, so that the shop sold 160 computers in 2001, 170 computer... show full transcript

Worked Solution & Example Answer:In the year 2000 a shop sold 150 computers - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 1

Step 1

Show that the shop sold 220 computers in 2007.

96%

114 rated

Answer

To find the number of computers sold in 2007, we can use the formula for the n-th term of an arithmetic sequence:

$n^{th} \text{ term} = a + (n - 1) d$

where:

$a = 150$ (the first term, number of computers sold in 2000)
$d = 10$ (the common difference, the number of additional computers sold each year)
$n = 8$ (2007 is the 8th term since we are starting from 2000)

Applying the formula:

$n^{th} \text{ term} = 150 + (8 - 1) \times 10$

$= 150 + 70 = 220$

Thus, the shop sold 220 computers in 2007.

Step 2

Calculate the total number of computers the shop sold from 2000 to 2013 inclusive.

99%

104 rated

Answer

The total number of computers sold can be found by calculating the sum of the arithmetic sequence from 2000 to 2013 inclusive (which consists of 14 terms). The formula for the sum $S_n$ of the first n terms is:

$S_n = \frac{n}{2} (a + l)$

where:

$n = 14$ (number of terms)
$a = 150$ (the first term)
$l = 150 + (14-1) \times 10 = 150 + 130 = 280$ (the last term)

Substituting these values into the sum formula:

$S_{14} = \frac{14}{2} (150 + 280) = 7 \times 430 = 3010$

Therefore, the total number of computers sold from 2000 to 2013 is 3010.

Step 3

In a particular year, the selling price of each computer in £s was equal to three times the number of computers the shop sold in that year.

96%

101 rated

Answer

Let the year in question be denoted as $y$ . The selling price in year $y$ can be expressed as:

$\text{Selling Price} = 900 - 20(y - 2000)$

The number of computers sold in year $y$ is:

$\text{Sold} = 150 + 10(y - 2000)$

According to the problem, we have:

$900 - 20(y - 2000) = 3(150 + 10(y - 2000))$

Expanding both sides gives:

$900 - 20y + 40000 = 450 + 30(y - 2000)$

Rearranging this leads to:

Which simplifies to:

Thus:

Therefore:

This results in the year 2009, confirming that this condition was met.

In the year 2000 a shop sold 150 computers - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 1

Worked Solution & Example Answer:In the year 2000 a shop sold 150 computers - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 1

Show that the shop sold 220 computers in 2007.

Calculate the total number of computers the shop sold from 2000 to 2013 inclusive.

In a particular year, the selling price of each computer in £s was equal to three times the number of computers the shop sold in that year.

Join the A-Level students using SimpleStudy...