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Sequences - nth Terms and Limiting Values

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 12 marks

In a linear sequence, the 1st term is 4.5

The 10th term is 27 more than the 1st term.

Work out an expression for the nth term.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Going from the 1st term to the 10th term adds the common difference 9 times, not 10.

Worked solution
  1. From the 1st term to the 10th term is 9 steps
  2. Common difference =27\div 9=3
  3. So the nth term starts 3n
  4. When n=1, 3n=3, but the 1st term is 4.5, so add 1.5
  5. Check: 10th term =30+1.5=31.5, which is 4.5+27 ✓
  6. Answer: 3n+1.5

Question 23 marks

A quadratic sequence starts

4, \quad 3, \quad -2, \quad -11, \quad \ldots

Work out an expression for its nth term.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Find the second difference and halve it to get the coefficient of n^2.

Worked solution
  1. First differences: -1, \ -5, \ -9
  2. Second difference: -4, so the n^2 term is -2n^2
  3. -2n^2 gives -2, \ -8, \ -18, \ -32
  4. Sequence minus -2n^2: 6, \ 11, \ 16, \ 21, which is 5n+1
  5. Check: n=3 gives -18+15+1=-2 ✓
  6. Answer: -2n^2+5n+1

Question 3Challenge6 marks

Sequence A has nth term \dfrac{4n+21}{n+3}

Sequence B has nth term 2n-7

(a)

Write down the limiting value of sequence A as n\to\infty

1 mark

(b)

There is one value of n for which the nth term of sequence A is equal to the nth term of sequence B.

Work out this value of n.

3 marks

(c)

How many terms of sequence A are greater than 4.2?

2 marks

Hint

For (b), set the two nth terms equal and clear the fraction; remember n must be a positive whole number.

Worked solution

Part (a)

  1. For large n, the 21 and the 3 hardly matter
  2. \dfrac{4n+21}{n+3}\approx\dfrac{4n}{n}=4
  3. Answer: 4

Part (b)

  1. Set the nth terms equal: \dfrac{4n+21}{n+3}=2n-7
  2. Multiply by (n+3): 4n+21=(2n-7)(n+3)=2n^2-n-21
  3. Rearrange: 2n^2-5n-42=0
  4. Factorise: (2n+7)(n-6)=0, so n=-3.5 or n=6
  5. n is a position, so it must be a positive whole number
  6. Answer: n=6 (both terms equal 5)

Part (c)

  1. Solve \dfrac{4n+21}{n+3}>4.2
  2. n+3 is positive, so 4n+21>4.2n+12.6
  3. 8.4>0.2n, so n<42
  4. The 42nd term equals exactly 4.2, which is not greater than 4.2
  5. The terms decrease towards 4, so terms 1 to 41 are greater than 4.2
  6. Answer: 41

More on this topic: Sequences - nth Terms and Limiting Values worksheet with full solutions

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