nth Term, Knowledge Organiser

Key Terms

• nth term: A formula giving any term in the sequence from its position n
• Common difference (d): The fixed amount added each term: becomes the coefficient of n
• Linear sequence: Sequence with a constant difference (nth term is dn + c)
• Position: The number of the term (1st, 2nd, 3rd…); use n in the formula
• Zero term: The value before the first term: helps find the constant c

Must-Know Facts

• The coefficient of n equals the common difference
• Find c: substitute n = 1 into dn and compare with the 1st term
• Or: find the "zero term" (term before the first), that's c
• To find a specific term: substitute the position number for n
• To check if a value is a term: set nth term = value and check n is a positive integer
• A decreasing sequence has a negative d (e.g. 10, 7, 4, 1 → nth term = −3n + 13)

Key Formulas

• nth term = dn + c
• d = common difference (any term minus the previous term)
• c = 1st term − d
• To find the 100th term: substitute n = 100 into the formula

Common Mistakes

• c = 1st term, not d: c is found by 1st term − d, not just the first term
• Wrong difference: Always check term₂ − term₁ = term₃ − term₂ before assuming linear sequence
• Checking membership: Set dn + c = target value: if n is not a positive integer, it is not in the sequence
• Confusing position and value: The 5th term means substitute n = 5, not find term with value 5
• Negative differences: Decreasing sequences have negative d: be careful with signs when finding c

Worked Examples

Find the nth term of the sequence: 5, 9, 13, 17, 21, ... Step 1 Find the common difference 9 − 5 = 4 13 − 9 = 4 Common difference d = 4 Step 2 Start the formula nth term starts with 4n Step 3 Compare to the sequence 4n gives: 4, 8, 12, 16, 20... Sequence is: 5, 9, 13, 17, 21... Each term is 1 more than 4n So add 1: 4n + 1 Step 4 Check it works n=1: 4(1)+1 = 5 ✓ n=2: 4(2)+1 = 9 ✓ n=3: 4(3)+1 = 13 ✓ Answer: nth term = 4n + 1

Worked Examples

The nth term of a sequence is 3n - 2 a) Find the 10th term b) Is 100 a term in this sequence? Part a Find the 10th term Substitute n = 10: 3(10) − 2 = 30 − 2 = 28 Part b Is 100 in the sequence? Set 3n − 2 = 100 and solve: 3n = 102 n = 34 Since n is a whole number, yes, 100 is the 34th term

Worked Examples

Find the nth term of: 20, 17, 14, 11, 8, ... Step 1 Find the common difference 17 − 20 = −3 Common difference d = −3 Step 2 Start the formula nth term starts with −3n Step 3 Compare and adjust −3n gives: −3, −6, −9, −12... Sequence is: 20, 17, 14, 11... 20 − (−3) = 23, so add 23 Answer: nth term = −3n + 23 (or 23 − 3n)