Sequences & Series Practice

What a sequence actually is

A sequence is a list of numbers arranged by position. The first number sits at position 1, the second at position 2, and the rule connecting a position to its number can be written as a formula for the nth term. Once that formula exists, any term can be found without writing the list out to the end.

Two families do most of the work in school. An arithmetic sequence adds the same amount at every step, called the common difference. A geometric sequence multiplies by the same amount at every step, called the common ratio. Telling them apart takes one subtraction and one division: check the gaps between terms, then check what each term is multiplied by. A sequence can continue forever, while a series is the sum of its terms. The list 3, 5, 7, ... is a sequence; 3 + 5 + 7 + ... is a series. Keeping those names separate helps when a question asks for a pattern or for a total.

What this ladder covers

Pattern-spotting begins the ladder, and a formula for any position ends it.

• Term-to-term rules: describe how to get from one term to the next, and use that rule to carry on.
• Position to term: find the number sitting at a stated position without listing everything before it.
• Linear nth terms: build a formula such as 2n + 1 from the common difference and the first term.
• Geometric sequences: spot the common ratio, and extend a sequence that grows by multiplication.
• Special lists: recognise square numbers and triangular numbers from their first few terms.
• Distant terms: substitute a large n into the formula, as in the 20th term of 3, 5, 7, ...

Worked examples

Find the nth term of 3, 5, 7, 9, ... The terms rise by 2, so the formula starts with 2n. For n = 1 that gives 2, which is one short of 3, so add 1: the nth term is 2n + 1. Checking the first term is what fixes the constant.

Find the nth term of 4, 7, 10, 13, ... The gap is 3, so start with 3n. At n = 1 that gives 3, and the first term is 4, so the formula is 3n + 1. Test n = 2: 3 × 2 + 1 = 7, which matches the list.

Find the nth term of 10, 8, 6, 4, ... The terms fall by 2, so the formula starts with −2n. At n = 1 that gives −2, and the first term is 10, so the constant is 12: the nth term is 12 − 2n. A decreasing sequence carries a negative coefficient.

Find the next term of 2, 6, 18, 54, ... Each term is three times the one before, so the next is 54 × 3 = 162. The gaps grow (4, 12, 36), which rules out an arithmetic rule straight away.

Find the 20th term of 3, 5, 7, ... The formula is 2n + 1, so substitute n = 20: 2 × 20 + 1 = 41. Counting up in steps of 2 from 3 nineteen times reaches the same number, but far more slowly.

Mistakes worth watching for

Counting is where nth-term answers usually unravel.

• Giving 3, 5, 7, ... the formula 2n + 3. The working was never checked at n = 1, where 2n + 3 gives 5 instead of 3. Substituting the first position catches this in seconds.
• Multiplying where the list only adds. 3, 6, 9, 12 rises by a gap of 3, so the next term is 15. Reading the gap as a multiplier gives 36 instead, and the list has changed family.
• Taking the 20th term as 3 + 2 × 20 = 43. There are 19 steps from the first term to the 20th, not 20, so the answer is 3 + 2 × 19 = 41. The formula 2n + 1 already handles the offset.
• Treating a geometric sequence as arithmetic. 2, 6, 18, 54 does not rise by a constant gap, so adding 4 to get the next term gives 58. Divide one term by the previous one to see the ratio of 3.
• Confusing the position with the value. n is the place in the list, never the number stored there. In 3, 5, 7 the term at position 3 is 7, so writing 3 × 3 = 9 mixes the place up with the value.
• Mixing up a sequence and a series. The terms 3, 5, 7 form a sequence; adding them gives the series total 15. A question about the 10th term asks for one entry, not the sum of the first ten.

How the ladder works

Each of the six rungs holds its own set of questions, five at the Warm-Up and eight or nine in the later ones. A set is cleared at 80% first-try accuracy, no rung starts locked, and retries generate fresh sequences.

Keep going

Produce a page of pattern questions with the sequences and series worksheet generator, check a distant term with the general calculator, or look at functions, where an nth term is one example of a rule taking an input to an output.

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