Nth Term Practice Test Worksheets
Nth term questions ask you to turn arithmetic sequences into a formula. This page shows how to spot the common difference, find the 0th term, and choose the correct expression for position n.
For students who can follow number patterns, but need faster and more accurate sequence formula practice.
Etsy listing opens in a new tab. Checkout and downloads are handled on Etsy.com.
What Do Nth Term Questions Look Like?
Students are given an arithmetic sequence as a list of numbers, a table of values, or a pattern problem and must find the general algebraic formula for position n. Questions test finding the common difference and calculating the 0th term adjustment. Here is a worked example.
Find an Increasing Sequence Rule
Question: Identify the formula for the nth term of this sequence:
1 3 5 7 9
Worked Method
Step 1: The sequence goes up by 2 each time, so the formula starts with 2n.
Step 2: Go back one step from the first term: 1 - 2 = -1.
Step 3: Combine the parts: 2n - 1.
Answer: C, 2n − 1.
Tip: Test n = 1. 2(1) - 1 = 1, so the formula matches the first term.
Rules to Help You Solve Nth Term Questions
As shown in the worked example above, finding an nth term formula relies on spotting the constant difference and stepping back to position zero. Use these key rules to solve increasing, table-based, and decreasing sequences systematically.
🔍 Rule 1: Find the common difference
Subtract one term from the next. This difference becomes the coefficient of n.
Example: 8, 11, 14, 17 has difference +3, so the formula starts 3n.
⬅ Rule 2: Step back to term 0
Position n = 1 is the first listed term. The 0th term is an imagined term one step before it, found by extending the same pattern backwards.
Term 0 = first term − common difference. For a decreasing sequence, subtracting a negative difference means adding: 46 − (−6) = 52.
Example: If term 1 is 8 and the difference is +3, term 0 is 5.
➕ Rule 3: Combine both parts
nth term = common difference × n + 0th term. Keep the sign of the 0th term: if it is −1, adding it gives 2n + (−1) = 2n − 1.
Example: Difference +3 and 0th term 5 gives 3n + 5.
⚠ Rule 4: Keep decreasing signs
For decreasing arithmetic sequences, the coefficient of n is negative. The imagined 0th term is always larger than term 1 because stepping backwards reverses the decrease.
Example: 46, 40, 34, 28 has difference -6 and 0th term 52, so 52 - 6n.
Two More Worked Examples
Practise extracting terms from a position-to-term table, then learn how to handle decreasing arithmetic sequences without falling into common sign traps.
Read a Sequence Table
Question: A sequence maps positions to terms as shown below. Which expression represents the nth term?
| Position (n) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Term | 8 | 11 | 14 | 17 | 20 |
Worked Method
Step 1: The terms increase by 3, so use 3n.
Step 2: The first term is 8. Step back by 3 to find term 0: 8 - 3 = 5.
Step 3: Add the 0th term to the n part: 3n + 5.
Answer: B, 3n + 5.
Tip: A table does not change the method. Use the term row to find the difference.
Handle a Decreasing Sequence
Question: Choose the correct expression for the nth term of the sequence:
46 40 34 28 22
Worked Method
Step 1: The sequence decreases by 6, so the formula starts with -6n.
Step 2: To go back one step before 46, add 6: 46 + 6 = 52.
Step 3: Combine the 0th term and the negative n part: 52 - 6n.
Answer: B, 52 − 6n.
Tip: Do not write 46 - 6n. That would give 40 when n = 1, not 46.
