0580

Sequences

Algebra and Sequences

Fractions and combinations

  • A sequence of fractions is usually two separate sequences stacked, so find the nth term of the tops and of the bottoms independently
    • In 1/3, 4/5, 9/7, 16/9 the tops are the square numbers and the bottoms go up in twos from 3, giving nth term n² ÷ (2n + 1)
  • Where a question defines one sequence in terms of two others, write out the terms of each, combine them, and then find the nth term of the result
  • It is often quicker to combine the two nth term expressions directly
    • If sequence A has nth term 4n − 1 and sequence B has nth term n², then the sequence formed by adding them has nth term n² + 4n − 1
  • Some questions give the terms as expressions in x and tell you the type of sequence, which turns into an equation to solve
    • For an exponential sequence, the ratio between the first pair equals the ratio between the second pair, and setting those equal gives the equation
Exam tip

The answer must be an expression in n, not the next term and not a description in words. For a linear sequence a part-correct formula still scores 1 of the 2 — the right multiple of n with a wrong constant, or the reverse. Check your formula on a term other than the first. These papers never use "common difference" or "common ratio"; they say "term-to-term rule" or "the nth term".

Worked example

A sequence of fractions

Find the nth term of 1/3, 2/5, 3/7, 4/9.

Solution:

  • Treat the numerators and the denominators as two separate sequences
  • Numerators: 1, 2, 3, 4 — a constant gap of 1 starting at 1, so the nth term is n
  • Denominators: 3, 5, 7, 9 — a constant gap of 2, so the formula contains 2n
  • 2n alone gives 2, 4, 6, 8, which is 1 short each time, so the denominator is 2n + 1
  • The nth term is n / (2n + 1)
  • Check at n = 4: 4 / (8 + 1) = 4/9, which matches