Mathematics · OCR H240 · Topic 1.04

Sequences and series

Binomial expansions, sequences, arithmetic and geometric series, and sigma notation.

  • 6 sub-topics
  • Taught in Year 13
  • 111 practice questions
  • 6 animated lessons
Practice questions (111) Lessons Worksheets coming soon All OCR Mathematics topics

What the specification says

Every sub-topic OCR can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. 1.04a-dBinomial expansionY13
  2. 1.04e-fSequencesY13
  3. 1.04gSigma notationY13
  4. 1.04hArithmetic sequencesY13
  5. 1.04i-jGeometric sequencesY13
  6. 1.04kModelling with sequences and seriesY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 111 questions in this topic, free, no account needed.

Sequences and series

Easy

Binomial expansion · Year 12 · 3 marks

  1. Expand (1+2x)4(1 + 2x)^4 fully.
    (3)
Show worked solutionHide worked solution
  1. Use row 4 of Pascal's triangle: 1, 4, 6, 4, 1.11+4(2x){}+ 4(2x)+6(2x)2{}+ 6(2x)^2+4(2x)3{}+ 4(2x)^3+(2x)4{}+ (2x)^4
  2. Simplify each term.=1+8x+24x2+32x3+16x4= 1 + 8x + 24x^2 + 32x^3 + 16x^4

Total3

  • M
  • A
  • B
How did it go?

Report a problem with this question

Sequences and series

Not needed for Year 12Medium

Geometric series · Year 13 · 4 marks

A geometric series has first term 12 and sum to infinity 48.
  1. (a)
    Find the common ratio.
    (2)
  2. (b)
    Find the sum of the first 10 terms, to 3 significant figures.
    (2)
Show worked solutionHide worked solution

Part (a)2 marks

  1. Use S∞=a1−rS_\infty = \dfrac{a}{1 - r}.121−r=48\dfrac{12}{1 - r} = 48
  2. Rearrange for rr. It is between −1 and 1, as it must be.1−r1 - r=14{}= \tfrac{1}{4}⇒r{}\Rightarrow r=34{}= \tfrac{3}{4}

Part (b)2 marks

  1. Use Sn=a(1−rn)1−rS_n = \dfrac{a(1 - r^n)}{1 - r}.S10S_{10}=12(1−0.7510)1−0.75{}= \dfrac{12\left(1 - 0.75^{10}\right)}{1 - 0.75}
  2. Round to 3 significant figures as asked.=45.3= 45.3

Total4

  • M
  • A
  • B
How did it go?

Report a problem with this question

Sequences and series

Not needed for Year 12Hard

Modelling population growth · Year 13 · 6 marks

A town had 2000 people at the start of 2024. The population is modelled as growing by 4% each year.
  1. (a)
    Write down an expression for the population nn years after the start of 2024.
    (1)
  2. (b)
    Find the first year in which the model predicts more than 3000 people.
    (4)
  3. (c)
    Give one reason why the model may not be reliable in the long term.
    (1)
Show worked solutionHide worked solution

Part (a)1 mark

  1. Growing by 4% means multiplying by 1.04 each year.2000×1.04n2000 \times 1.04^n

Part (b)4 marks

  1. 2000×1.04n>30002000 \times 1.04^n > 3000⇒1.04n>1.5{}\Rightarrow 1.04^n > 1.5
  2. Take logs of both sides.n>ln⁡1.5ln⁡1.04n > \dfrac{\ln 1.5}{\ln 1.04}=10.34{}= 10.34
  3. The first whole number of years after 10.34.n=11n = 11, so during 2035

Part (c)1 mark

  1. Any sensible reason.Growth cannot carry on for ever: housing and space are limited

Total6

  • M
  • A
  • B
How did it go?

Report a problem with this question