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A geometric series has first term a and common ratio r - Edexcel - A-Level Maths Pure - Question 1 - 2005 - Paper 2
Question 1
A geometric series has first term a and common ratio r. Prove that the sum of the first n terms of the series is \[ S_n = \frac{a(1 - r^n)}{1 - r} \] Mr King will b... show full transcript
Worked Solution & Example Answer:A geometric series has first term a and common ratio r - Edexcel - A-Level Maths Pure - Question 1 - 2005 - Paper 2
Step 1
Prove that the sum of the first n terms of the series is \[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Answer
To prove this, we can consider the sum of the first n terms of a geometric series:
[ S_n = a + ar + ar^2 + ... + ar^{n-1} ]
Multiplying both sides by the common ratio, r, gives us:
[ rS_n = ar + ar^2 + ar^3 + ... + ar^n ]
Now, subtract the second equation from the first:
[ S_n - rS_n = a - ar^n ]
This simplifies to:
[ S_n(1 - r) = a(1 - r^n) ]
Finally, dividing by (1 - r) yields the required formula:
[ S_n = \frac{a(1 - r^n)}{1 - r} ]
Step 2
Find, to the nearest £100, Mr King’s salary in the year 2008.
Answer
Mr King's salary forms a geometric sequence with a first term of £35,000, and a common ratio of 1.04 (corresponding to the 4% increase).
To find his salary in 2008, we note that this is the 4th term (since 2005 is the first year):
[ S_4 = a r^{n-1} = 35000 \times (1.04)^{3} \approx 39400 ]
Therefore, to the nearest £100, Mr King’s salary in 2008 is £39,400.
Step 3
Find, to the nearest £100, the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024.
Answer
Mr King's retiring year is 2024, meaning he will receive salary from 2005 to 2024, a total of 20 years.
Using the formula for the sum of a geometric series:
[ S_n = \frac{a(1 - r^n)}{1 - r} ]
where:
- a = £35,000
- r = 1.04
- n = 20
Thus,
[ S_{20} = \frac{35000(1 - (1.04)^{20})}{1 - 1.04} ]
Calculating this gives approximately:
[ S_{20} \approx 1042000 ]
So the total salary from 2005 until 2024 is £1,042,000, to the nearest £100.