nth term of sequences

Sequences appear on all OCR J560 papers. The nth term formula allows any term to be found directly. Linear nth terms are Foundation and Higher; quadratic nth terms (an² + bn + c) are Higher-tier only.

Linear (arithmetic) sequences

A linear sequence has a constant difference between terms.

nth term = dn + c where d = common difference.

Method:

1. Find the common difference d (nth term coefficient).
2. Write dn. Compare the sequence with dn to find c.

Example: 5, 8, 11, 14, ...

• Difference = 3 → coefficient is 3.
• Sequence of 3n: 3, 6, 9, 12, ...
• The given sequence is 2 more than 3n at each position.
• nth term = 3n + 2.
• Check: n=1 → 5 ✓; n=4 → 14 ✓.

Example: 20, 17, 14, 11, ...

• Difference = −3 → coefficient is −3.
• −3n gives −3, −6, −9, −12, ...
• Difference from sequence: 20−(−3)=23.
• nth term = −3n + 23 (or 23 − 3n).

Finding a specific term

Substitute n into the nth term formula.

Example: nth term = 4n − 1. Find the 20th term: 4(20) − 1 = 79.

Is a value in the sequence?

Set the nth term equal to the value and solve. If n is a positive integer, it's in the sequence.

Example: Is 97 in the sequence 3n + 2? 3n + 2 = 97 → 3n = 95 → n = 31.67... → NOT an integer → 97 is not in the sequence.

Quadratic sequences

A quadratic sequence has a constant second difference.

nth term = an² + bn + c.

Method:

1. Find first differences, then second differences.
2. a = (second difference) ÷ 2.
3. Form an², subtract from the sequence to get a linear remainder.
4. Find the nth term of the remainder (linear method above).

Example: 3, 10, 21, 36, 55, ...

• 1st differences: 7, 11, 15, 19 (increasing by 4).
• 2nd differences: 4, 4, 4 → constant → quadratic confirmed.
• a = 4 ÷ 2 = 2.
• Subtract 2n² from sequence: 3−2=1, 10−8=2, 21−18=3, 36−32=4, 55−50=5 → sequence 1,2,3,4,5 → linear nth term = n.
• Full nth term = 2n² + n.
• Check: n=1 → 2+1=3 ✓; n=3 → 18+3=21 ✓.

Common OCR exam mistakes

1. Confusing the nth term with the "difference" — the nth term is a FORMULA, not just the difference.
2. Not checking the answer by substituting n=1 and n=2 (at minimum).
3. Quadratic: dividing the FIRST difference by 2 instead of the SECOND difference by 2.
4. Finding which term has a given value: forgetting to check n is a positive integer.