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A geometric series is a + ar + ar² + .. - Edexcel - A-Level Maths Pure - Question 2 - 2012 - Paper 4
Question 2
A geometric series is a + ar + ar² + ... (a) Prove that the sum of the first n terms of this series is given by Sₙ = \( \frac{a(1 - r^n)}{1 - r} \) The third and... show full transcript
Worked Solution & Example Answer:A geometric series is a + ar + ar² + .. - Edexcel - A-Level Maths Pure - Question 2 - 2012 - Paper 4
Step 1
Prove that the sum of the first n terms of this series is given by Sₙ = \( \frac{a(1 - r^n)}{1 - r} \)
Answer
To prove this formula, we start with the geometric series:
Sₙ = a + ar + ar² + ... + ar^{n-1}.
Multiplying the entire equation by (1 - r), we get:
( Sₙ(1 - r) = a(1 - r) + ar + ar² + ... + ar^{n-1}(1 - r) )
The right-hand side simplifies to:
( Sₙ(1 - r) = a(1 - r) + ar^n - ar^n ).
So, we can deduce:
( Sₙ(1 - r) = a - ar^n ).
Now, rearranging gives: ( Sₙ = \frac{a(1 - r^n)}{1 - r} ).
Step 2
the common ratio
Answer
Given the third term ( ar^2 = 5.4 ) and the fifth term ( ar^4 = 1.944 ), we can form the equation:
[ \frac{ar^4}{ar^2} = \frac{1.944}{5.4} ]
This simplifies to:
[ r^2 = \frac{1.944}{5.4} ]
Calculating the right-hand side gives:
[ r^2 = 0.36 \implies r = \sqrt{0.36} = 0.6. ]
Step 3
Step 4