Sequences & Series Worksheets

What this worksheet covers

This pattern sheet is generated from the same question rules used in the practice ladder. Its optional key can print separately, with the working for each item rather than a fixed bank of answers.

A sheet can ask any of these:

• find the nth term of 5, 8, 11, 14
• find the nth term of 20, 17, 14, 11
• find the 15th term of 5, 8, 11
• find the next term of 4, 12, 36
• find the nth term of 2, 5, 8, 11
• is 101 a term of 5, 8, 11
• write the first four terms of 4n − 1

There is nothing to shade in and no graph paper needed, so a full set prints cleanly on one side.

Choosing a level

The level progression shifts from spotting the next term to describing a rule. Warm-Up continues a short list; Basics finds an nth term for a rising pattern, Practice handles a falling one, and Steady mixes both signs. Challenge uses multiplication between terms, while Mastery tests whether a target number appears in the sequence.

Choose a whole-number count of 3–15. Keep the selected rung for every item, or begin a Mixed difficulties run there and let it cycle through later stages. A spiral sheet can mix this topic with another, but it needs at least two topics selected.

Write-in leaves room to show how the rule was formed. A–D works for a quick retrieval check, since each response is a short expression or one number.

A worked example

For each linear rule, substitute the first position to check that it returns the list's opening value.

Find the nth term of 5, 8, 11, 14. The jump from one term to the next is 3, so the rule carries 3n. At n = 1 the rule has to give 5, and 3n alone gives 3, so 2 is added: the nth term is 3n + 2. Checking n = 1 returns 5, which is the first term printed on the sheet.

Find the nth term of 20, 17, 14, 11. The terms fall by 3, so the rule reads 23 − 3n. Writing 20 − 3n would give 17 at n = 1, one term too far along. Checking n = 1 returns 20.

Marking with the answer key

A formula can match the step size and still miss every listed value if its starting offset is wrong.

• The nth term of 5, 8, 11 given as 3n + 5. The 5 was dropped in as the constant, which returns 8 at n = 1. The rule is 3n + 2.
• 20, 17, 14 given as 20 − 3n. That fits n = 0, so every value lands one place late: n = 1 gives 17, not 20. The nth term is 23 − 3n.
• The 15th term of 5, 8, 11 written as 3 × 15 + 5 = 50. The constant is 2, not 5, so the working is 3 × 15 + 2 = 47.
• The next term of 4, 12, 36 written as 60. That adds 24, which happens to fit the last jump. The sequence multiplies by 3, so the next term is 108.

Printing and using it

Pairing a number pattern with a straight-line graph makes the shared rule visible. If learners have just plotted y = 3x + 2, they can compare its output with the matching nth-term rule 3n + 2.

Standard gives each task a line for its rule and another for checking the first term. Large-print gives the subscript more room; Compact fits twelve prompts on one side.

Put the learner's name, date and class in the header when the page will be collected. Dyslexia-friendly type separates the subscript from its letter, helpful when a₁ and n appear together. Stretch gives each third prompt a harder variation, and is only available when all questions use one difficulty. Rename the heading to identify the pattern, for example "Market stall sequences".

For a partner activity, cut four terms from an extra copy and shuffle them. Put the cards in order, identify the step, then write a rule that fits the list.

Keep going

Take the same patterns onto the sequences and series practice ladder, which marks each answer as you go, read the lesson on sequences and series for sigma notation, or generate functions worksheets, where the same rule is written with x instead of n.

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