What is the 'Nth-term rule' in linear (arithmetic) sequences and how is it used?

In any sequence, the position of a term (1st, 2nd, 3rd etc.) is represented by the letter ‘n’. In linear sequences only, the ‘nth-term rule’ gives the value of any term in that sequence at position ‘n’. It is written as ‘xn ± y’ where x = the constant difference between term values and y is a particular number. The rule allows you to work out the terms of a sequence. You will be required to both write a sequence using it and find it from a sequence. Examples:

I) Writing a sequence from the nth-term rule: Q: Write the first 5 terms of the sequence 20 - 4n.

A: To work out the value of a term, substitute the position for ‘n’. E.g For the 1st term: n = 1. Therefore: 20 - 4n ⇒ 20 - 4(1) ⇒ 20 - (4 x 1) = 20 - 4 = 16. For 2nd term: n = 2. Therefore: 20 - 4n ⇒ 20 - 4(2) ⇒ 20 - (4 x 2) = 20 - 8 = 12. Solve for the remaining terms.

II) Finding the nth-term rule from a sequence: Q: What is the nth-term rule for the linear sequence: -5, -2, 1, 4, 7 … ?

A: Determine the common difference between the terms: 2nd - 1st ⇒ -2 - -5 = 3, Common difference: +3. Given the formula for the nth-term rule is ‘xn ± y’ as explained above, the nth-term rule for this sequence is 3n ± y. To find ‘y’, substitute in the term and corresponding value and solve: E.g Using the 1st term: 3n ± y = -5, 3(1) ± y = -5, 3 ± y = -5, y = -5 - 3 ⇒ y = -8. Cross check with other terms.