Sequences and series: watch it solved step by step

Spot whether a sequence is arithmetic, geometric or quadratic, then find its n-th term and the sum of the first n terms. Level: GCSE / A-level. Type any problem of this kind and the Lab works it out and animates every move, with a plain-words explanation of why.

Worked example: 2, 6, 18, 54, ... n = 8

  1. The sequence. A sequence is a list of numbers that follows a rule. The first job is to spot the rule.
  2. Look at the ratios. The differences are not constant, so try dividing each term by the one before. If the answer is always the same, the sequence is geometric: it is multiplied by a fixed amount each time.
  3. Write the n-th term formula. Start at a and multiply by the common ratio r a total of n-1 times.
  4. Find the 8th term. Put n = 8 into the formula.
  5. Sum of the first 8 terms. For a geometric sequence the sum of n terms is a(r^n - 1)/(r - 1).
  6. Check. Add the terms up one by one and compare with the formula.

Answer: geometric: uₙ = 2 × (3)^(n − 1); u8 = 4374; S8 = 6560

Practise sequences and series with problems the Lab marks for you.

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