Sequences and nth terms

A position and a value are different numbers

For the sequence 5, 8, 11, 14, the positions are 1, 2, 3, 4. Each step adds 3. Three times the position gives 3, 6, 9, 12; adding 2 gives the required terms. A rule for this arithmetic sequence is uₙ = 3n + 2 for positive integer n.

The 20th term is 3 × 20 + 2 = 62. To ask which term is 80, solve 3n + 2 = 80, giving n = 26. To ask whether 81 occurs, solve n = 79/3, which is not a positive integer, so 81 is not a term under this rule.

Build the rule from a structure

A row of adjacent squares made from matchsticks uses 4 sticks for one square. Each extra square shares one side and adds 3 sticks. The count is 4 + 3(n-1) = 3n + 1.

The first term matters: 3n would start at 3 and miss the extra end stick. Drawing the first few arrangements checks what the variable counts.

A finite list does not prove the continuation

The terms 2, 4, 6 fit the rule 2n, but other rules can share those first three terms and differ later. A problem must state the intended pattern, supply a geometric construction or restrict the type of rule. “The next term must be 8” needs such an assumption.

Check your understanding 1 A common difference is not the first term

An arithmetic sequence starts 7, 12, 17, 22. Find its nth term and decide whether 102 is in the sequence.

Your answer or drawing on paper

Show answer

The difference is 5, and 5n + 2 gives the first term 7. Solve 5n + 2 = 102 to get n = 20, so 102 is the 20th term.

Reverse the relationship

A table says uₙ = 2n + 1. A learner substitutes n = 11 to answer “Which term equals 11?” Correct the decision.