Most tested M4.19

Finding the Nth Term

This topic involves identifying the underlying algebraic rule (the 'nth term') for a given list of numbers. It's a key skill for recognising and describing mathematical patterns, which is fundamental in many engineering and scientific contexts.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • A linear sequence has a constant difference between consecutive terms; its nth term is of the form an + b.
  • The coefficient 'a' in a linear sequence's nth term is equal to this constant difference.
  • A quadratic sequence has a constant second difference (the difference between the differences); its nth term is of the form an2 + bn + c.
  • The coefficient 'a' of the n2 term in a quadratic sequence is always half the value of the constant second difference.
  • Once the 'an2' part is found, subtract it from the original sequence to reveal a new, simpler linear sequence. Finding the nth term of this residual sequence gives you 'bn + c'.

Formulae

Tn = an + b

For a linear sequence, where 'a' is the common difference and 'b' is the 'zeroth term' (the value before the first term).

Tn = an2 + bn + c

For a quadratic sequence, where 'a' is half the constant second difference.

Definitions

Linear Sequence
A sequence of numbers where the difference between any two consecutive terms is the same. Also known as an arithmetic progression.
Quadratic Sequence
A sequence of numbers where the first differences are not constant, but the second differences are.
nth term
An algebraic formula that allows you to calculate any term in a sequence by substituting its position number, n.

Worked example

A sequence begins with the terms -1, 8, 23, 44, ⋯ Find an expression for the nth term of this sequence.

  1. 1

    Calculate the first differences between terms:

    8 - (-1) = 9; 23 - 8 = 15; 44 - 23 = 21

    The first differences are 9, 15, 21.

  2. 2

    Since the first differences are not constant, calculate the second differences:

    15 - 9 = 6; 21 - 15 = 6

    The second difference is a constant +6.

  3. 3

    The sequence is quadratic.

    The coefficient of n2 is half the second difference:

    a = 6 / 2 = 3

    The rule starts with 3n2.

  4. 4

    Write down the original sequence and the sequence for 3n2 (for n=1, 2, 3, 4):

    Original:

    -1, 8, 23, 44 3n2:

    3(12)=3, 3(22)=12, 3(32)=27, 3(42)=48
  5. 5

    Subtract the 3n2 sequence from the original sequence to find the residual:

    -1 - 3 = -4 8 - 12 = -4 23 - 27 = -4 44 - 48 = -4
  6. 6

    The residual sequence is a constant -4.

    This means the 'bn + c' part is simply -4.

  7. 7

    Combine the two parts to get the final nth term.

Answer: 3n2 - 4

Common mistakes

  • ×Forgetting to divide the constant second difference by two when finding the 'a' coefficient for a quadratic sequence.
  • ×Making sign errors, especially when dealing with decreasing sequences or when subtracting a larger term from a smaller term to find the residual.
  • ×Arithmetic mistakes when calculating the first or second differences, which causes all subsequent steps to be incorrect.

No-calculator tips

  • ✓For a linear sequence an + b, find the common difference 'a', then work backwards one step from the first term to find the 'zeroth term', which is 'b'. For 7, 10, 13⋯, a=+3. The zeroth term is 7-3=4, so the rule is 3n+4.
  • ✓When finding the residual sequence for a quadratic, write the original sequence directly above the 'an2' sequence and perform the subtraction vertically for each term to minimise errors.
  • ✓Always check your final nth term formula by plugging in n=1 and n=2 to see if it generates the first two terms of the original sequence correctly.

Read this topic in the official UAT-UK ESAT guide →

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