9. Jess started work 20 years ago. In year 1 her annual salary was £17000. Her annual salary increased by £1500 each year, so that her annual salary in year 2 was £18500, in year 3 it was £20000 and so on, forming an arithmetic sequence. This continued until she reached her maximum annual salary of £32000 in year k. Her annual salary then remained at £32000. (a) Find the value of the constant k. (b) Calculate the total amount that Jess has earned in the 20 years.

A-Level Edexcel Maths Pure Rational Expressions: 9. Jess started work 20 years ag

9. Jess started work 20 years ago - Edexcel - A-Level Maths Pure - Question 1 - 2015 - Paper 1

Question 1

9. Jess started work 20 years ago. In year 1 her annual salary was £17000. Her annual salary increased by £1500 each year, so that her annual salary in year 2 was £1... show full transcript

Worked Solution & Example Answer:9. Jess started work 20 years ago - Edexcel - A-Level Maths Pure - Question 1 - 2015 - Paper 1

Step 1

Find the value of the constant k.

96%

114 rated

Answer

To find the value of k, we first determine how many years it takes for Jess's salary to reach £32000 starting from £17000 with an annual increase of £1500.

Establish the arithmetic sequence:
- Year 1: £17000
- Year 2: £18500
- Year 3: £20000
- So, the salary in year n can be expressed as: $S_n = 17000 + (n - 1) \times 1500$
Set the equation to find k:
- We need to find k such that: $17000 + (k - 1) \times 1500 = 32000$
- Rearranging gives: $(k - 1) \times 1500 = 32000 - 17000$ $(k - 1) \times 1500 = 15000$
- Dividing both sides by 1500: $k - 1 = 10$ $k = 11$

Thus, the value of the constant k is 11.

Step 2

Calculate the total amount that Jess has earned in the 20 years.

99%

104 rated

Answer

To calculate the total amount Jess has earned over 20 years, we will consider the first 11 years where her salary increases, and the last 9 years where she earns a fixed salary of £32000.

Total Salary for the first 11 years:
- Year 1: £17000
- Year 2: £18500
- Year 3: £20000
- Year 4: £21500
- Year 5: £23000
- Year 6: £24500
- Year 7: £26000
- Year 8: £27500
- Year 9: £29000
- Year 10: £30500
- Year 11: £32000
The total for these 11 years can be calculated as: $ext{Total}_{11} = 17000 + 18500 + 20000 + 21500 + 23000 + 24500 + 26000 + 27500 + 29000 + 30500 + 32000$ This is an arithmetic series where:
- First term (a) = £17000
- Last term (l) = £32000
- Number of terms (n) = 11
- Using the formula for the sum of an arithmetic series: $S_n = \frac{n}{2} (a + l)$ $S_{11} = \frac{11}{2} (17000 + 32000) = \frac{11}{2} \times 49000 = 269500$
Total Salary for the last 9 years:
- For year 12 to 20, Jess earns a fixed salary of £32000.
- Total for these 9 years: $ext{Total}_{9} = 9 \times 32000 = 288000$
Combine the totals:
- Total earnings over 20 years: $ext{Total}_{20} = ext{Total}_{11} + ext{Total}_{9} = 269500 + 288000 = 557500$

Thus, the total amount that Jess has earned in 20 years is £557500.

9. Jess started work 20 years ago - Edexcel - A-Level Maths Pure - Question 1 - 2015 - Paper 1

Worked Solution & Example Answer:9. Jess started work 20 years ago - Edexcel - A-Level Maths Pure - Question 1 - 2015 - Paper 1

Find the value of the constant k.

Calculate the total amount that Jess has earned in the 20 years.

Join the A-Level students using SimpleStudy...