Sequence Sleuth

Sequence Sleuth

A sequence is a list of numbers following a rule, and your job is to uncover it. Crack the rule and you can continue the list for ever, or jump straight to the 100th term without writing out the 99 before it. The term-to-term rule is the first clue, and it comes in two kinds:

What is the rule?

Look at the sequence 1, 5, 25, 125, ... What is its term-to-term rule?

• Multiply by 5
• Add 4
• Add the two previous terms
• Multiply by 4

Find the next term

A sequence goes 5, 9, 13, 17, ... What is the NEXT term?

The nth-term formula

The term-to-term rule only ever takes you one step forward. The nth term takes you straight to *any* position n. For a linear (arithmetic) sequence: nth term = (common difference) × n + (first term − common difference) For 3, 5, 7, 9 the common difference is 2, and 3 − 2 = 1, so the nth term is 2n + 1. Always check it: n = 1 gives 2(1) + 1 = 3, which is the first term. ✓ The second part catches people out. The difference alone gives you 2n, which would be 2, 4, 6, 8 - the right steps but the wrong starting point. The adjustment slides the whole sequence into place.

Order the method

Put the steps for finding the nth term of any linear sequence into the right order.

• Find the common difference between consecutive terms
• Write that difference in front of n
• Work out the first term minus the common difference
• Add that adjustment on the end, and check it gives the first term when n = 1

The coefficient of n

A sequence goes 4, 7, 10, 13, ... In its nth term, what number multiplies n?

Build the nth term

To find the nth term of 5, 9, 13, 17, first work out the common difference: 9 − 5 = _____. That number goes in front of n. Now the adjustment: first term minus the common difference = 5 − 4 = _____. So the nth term is _____. Checking with n = 1 gives _____, which is the first term.

Jump to the 10th term

Use the nth term you just built for 5, 9, 13, 17 to find the 10th term of that sequence.

Is that number in the sequence?

The nth term does one more job, and it is the one examiners like best. Is 100 a term of 3, 5, 7, 9, ...? 1. The nth term is 2n + 1, so set it equal to the number: 2n + 1 = 100. 2. Solve for n: 2n = 99, so n = 49.5. 3. A position has to be a whole number - there is no 49.5th term - so 100 is not in the sequence. Had n come out as a whole number, that number would be the position where it sits. This is why you never answer a membership question by writing the sequence out and hoping.

In or out?

The sequence 5, 9, 13, 17, ... has nth term 4n + 1. Is 42 a term of this sequence?

• No: 4n + 1 = 42 gives n = 10.25, not a whole position
• Yes: it is the 10th term of the sequence
• Yes: it is the 11th term of the sequence
• No: the sequence contains only multiples of 4

Sequences on a graph

Plot a linear sequence with the position n across and the term up, and the points lie on a straight line. That is exactly what "linear" means. The common difference is the line's gradient: a sequence going up in 3s climbs 3 for every 1 step across. This is the same fact as the nth term, drawn instead of written, and it is a quick check on your arithmetic: a point that does not sit on the line is a term you got wrong.

Plot the term

The sequence 4, 7, 10, 13, ... is plotted with position n across and the term up. Work out the 5th term, then plot the point for it.

Match each special sequence to its name