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Sequences Worksheets

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Easy

10 problems

Medium

20 problems

Hard

20 problems

Mixed

30 problems

Free printable sequences worksheets with step-by-step answer keys. Every worksheet is uniquely generated so students never see the same problems twice. Topics covered range from continue arithmetic sequence at the easy level through to sum of arithmetic series at the advanced level.

CCSS.HSF.BFCCSS.HSF.LE

What is sequences?

A sequence is an ordered list of numbers that follow a specific pattern or rule. Arithmetic sequences increase by a constant difference between consecutive terms, while geometric sequences multiply by a constant ratio. The sequence 2, 5, 8, 11 adds 3 each time, making it arithmetic with a common difference of 3.

Why it matters

Sequences appear throughout mathematics and real-world applications. In finance, compound interest follows geometric sequences where money grows by a constant percentage each year — $1000 at 5% annual interest becomes $1050, $1102.50, $1157.63. Population growth models use geometric sequences to predict changes over time. In physics, objects in free fall follow arithmetic sequences for velocity — dropping 9.8 m/s faster each second. Sequences form the foundation for calculus series and appear in computer algorithms. Architecture uses arithmetic sequences in step designs and geometric sequences in spiral structures. Understanding sequences is essential for CCSS.HSF.BF and CCSS.HSF.LE standards, preparing students for advanced topics like limits, derivatives, and mathematical modeling in college-level courses.

Common mistakes to watch for

  • ✗Confusing arithmetic and geometric patterns leads to errors like treating 2, 6, 18, 54 as arithmetic (difference of 4, 12, 36) instead of geometric (ratio of 3).
  • ✗Incorrect formula application produces wrong answers such as calculating the 8th term of 3, 7, 11, 15 as 3 + 8 × 4 = 35 instead of 3 + (8-1) × 4 = 31.
  • ✗Misidentifying the first term results in errors like using a₁ = 5 for the sequence 2, 5, 8, 11 instead of a₁ = 2, leading to incorrect calculations.

Questions teachers ask

What is the difference between arithmetic and geometric sequences?+
Arithmetic sequences add the same number each time (like 3, 7, 11, 15 adding 4), while geometric sequences multiply by the same number each time (like 2, 6, 18, 54 multiplying by 3). Check differences first, then ratios to identify the type.
How do you find the nth term of an arithmetic sequence?+
Use the formula aₙ = a₁ + (n-1)d where a₁ is the first term, n is the position, and d is the common difference. For sequence 5, 9, 13, 17, the 10th term is 5 + (10-1) × 4 = 41.
How do you identify if a sequence is arithmetic or geometric?+
Calculate differences between consecutive terms first. If they're equal, it's arithmetic. If not, calculate ratios by dividing each term by the previous one. Equal ratios indicate a geometric sequence. Some sequences are neither type.
What is the sum formula for arithmetic series?+
The sum of n terms in an arithmetic series is S = n/2 × (first term + last term) or S = n/2 × (2a₁ + (n-1)d). For the first 10 terms of 3, 7, 11, 15, the sum is 10/2 × (3 + 39) = 210.
Can a sequence have negative terms or fractions?+
Yes, sequences can contain negative numbers, fractions, or decimals. The sequence -5, -2, 1, 4 is arithmetic with d = 3. The sequence 1/2, 1, 2, 4 is geometric with r = 2. The pattern rules remain the same regardless of number type.
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Beginner

Generate →
Concepts
Continue arithmetic sequence
Range
d: 2–10, start 1–10
Steps
1 step
Example
2, 5, 8, __, __, __

Easy

Generate →
Concepts
Find nth term using formula
Range
a₁: 1–10, d: 2–6, n=8–15
Steps
2–3 steps
Example
Find 10th term of 3, 7, 11, …

Medium

Generate →
Concepts
Find common difference and nth term
Range
a₁: 1–10, d: 3–9, n=20
Steps
3 steps
Example
Find d and 20th term

Hard

Generate →
Concepts
Sum of arithmetic series
Range
a₁: 1–5, d: 2–4, n=10–15
Steps
3–4 steps
Example
Sum of first 10 terms

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