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4MA1/1HR Jan 2019, Question 12: Expand and Simplify

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 5 marks

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The question, in short

From the fraction sequence 1/1, 2/3, 3/5, 4/7, find a formula for the nth fraction and show algebraically that an odd number squared leaves remainder 1 when divided by 4.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 2 marks

Give the nth term of the fraction sequence in terms of n, using the numerator and denominator patterns.

  1. 1Numerator is n
  2. 2Denominator is the nth odd number, which is 2n − 1
  3. 3So the nth term is n / (2n − 1)

Answern/(2n-1)

Part (b) · 3 marks

Use algebra to demonstrate that squaring any odd number and dividing by 4 always gives remainder 1.

  1. 1Any odd number can be written as 2n − 1, where n is an integer
  2. 2(2n − 1)² = 4n² − 4n + 1
  3. 3= 4(n² − n) + 1, which is a multiple of 4 plus 1
  4. 4So (2n − 1)² = 4(n² − n) + 1, a multiple of 4 plus 1, so the remainder is 1

Answer(2n − 1)² = 4n² − 4n + 1 = 4(n² − n) + 1, a multiple of 4 plus 1, so the remainder is 1

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