A girl saves money over a period of 200 weeks. She saves 5p in Week 1, 7p in Week 2, 9p in Week 3, and so on until Week 200. Her weekly savings form an arithmetic sequence. (a) Find the amount she saves in Week 200. (b) Calculate her total savings over the complete 200 week period.

A-Level Edexcel Maths Pure Simultaneous Equations: A girl saves money over a period

A girl saves money over a period of 200 weeks - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 1

Question 6

A girl saves money over a period of 200 weeks. She saves 5p in Week 1, 7p in Week 2, 9p in Week 3, and so on until Week 200. Her weekly savings form an arithmetic se... show full transcript

Worked Solution & Example Answer:A girl saves money over a period of 200 weeks - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 1

Step 1

Find the amount she saves in Week 200.

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Answer

To determine the amount saved in Week 200, we first recognize the arithmetic sequence of weekly savings. The first term (Week 1) is given as 5p, and the common difference can be calculated based on the changes in weekly savings:

Week 1: 5p
Week 2: 7p
Week 3: 9p

From the above, we see that each week she saves an additional 2p.

Now, using the formula for the nth term of an arithmetic sequence: $a_n = a + (n-1)d$ where:

$a = 5p$ (the first term),
$d = 2p$ (the common difference), and
$n = 200$ (the term we want).

Substituting these values into the formula: $a_{200} = 5p + (200 - 1)(2p)$ $a_{200} = 5p + 199 imes 2p$ $a_{200} = 5p + 398p = 403p$

Thus, the amount saved in Week 200 is 403p.

Step 2

Calculate her total savings over the complete 200 week period.

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Answer

The total savings can be calculated using the formula for the sum of an arithmetic series:

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

$S_n$ is the total sum,
$n$ is the total number of terms (200 weeks),
$a$ is the first term (5p), and
$l$ is the last term, which we have already calculated as 403p.

Substituting the values: $S_{200} = \frac{200}{2} (5p + 403p)$ $S_{200} = 100 (408p)$ $S_{200} = 40800p$

Therefore, the total savings over the complete 200 week period is 40800p.

A girl saves money over a period of 200 weeks - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 1

Worked Solution & Example Answer:A girl saves money over a period of 200 weeks - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 1

Find the amount she saves in Week 200.

Calculate her total savings over the complete 200 week period.

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