Make the notation concrete
In tn, the subscript labels a term; it does not mean multiplication by n. A recurrence tells you how to obtain another term from earlier ones. A formula for the nth term and a formula for the sum answer different questions.
- Write the initial term and calculate the next few terms directly.
- Look at differences or ratios. If neither is constant, try subtracting a fixed value.
- Check any proposed formula against the initial term and the recurrence, not just a few numerical examples.
Worked example
A recurrence hiding a geometric sequence
Let t1 = 2 and tn+1 = 3tn − 2 for n ≥ 1. Find a formula for tn.
The first four terms are 2, 4, 10 and 28. Their ratios are not constant. The fixed value satisfies c = 3c − 2, giving c = 1.
Subtract 1 from both sides of the recurrence:
tn+1 − 1 = 3(tn − 1).
Thus un = tn − 1 is geometric, with first term 1 and ratio 3. Hence tn = 1 + 3n−1.
At n = 1 this gives 2. Substituting into 3tn − 2 gives 1 + 3n, exactly the proposed formula for tn+1. This verifies the formula for the sequence.
Your turn
A sum with cancellation
For a positive integer n, find 1/(1 × 2) + 1/(2 × 3) + ··· + 1/[n(n + 1)].
Show a hint
Write 1/[k(k + 1)] as a difference of two simple fractions.
Show the full solution
Since 1/k − 1/(k + 1) = 1/[k(k + 1)], the sum becomes
(1 − 1/2) + (1/2 − 1/3) + ··· + (1/n − 1/(n + 1)).
All the internal terms cancel. The result is 1 − 1/(n + 1) = n/(n + 1).
For n = 1 this is 1/2, matching the first term. As n increases, the sum approaches 1 from below, which is consistent with adding positive terms.
Write out the n = 3 case and point to every cancellation. Then explain why the same pattern holds for any positive integer n.
What to practise next
Practise recurrence notation and geometric series separately. Then mix recurrence questions with sums so that you must decide which method applies.
- Physics & Maths Tutor: sequences and series
Topic questions and model answers for sequences and sums.
- STEP archive and support resources
For a longer challenge, use the linked STEP question database to look for sequence and series problems.
The examples and explanations on this page are part of Arij Asad’s teaching guides. Linked papers and worksheets belong to their named creators and publishers. PMT’s topic collections include exam-board questions and other credited materials.