MathematicsExam code: 0580

Sequences

Algebra and Sequences

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

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

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