Sequences and Series Worksheets

What are sequences and series?

A sequence is an ordered list of numbers: $a_{1}, a_{2}, a_{3}, \dots$. In an arithmetic sequence the same number $d$ (the common difference) is added each time. In a geometric sequence each term is multiplied by the same number $r$ (the common ratio). A series is the sum of the terms of a sequence.

The formulas used on these worksheets are:

• Easy — find a term of an arithmetic sequence. The difference can be negative.
• Medium — find a term of a geometric sequence with ratio 2, 3 or −2.
• Hard — find the sum $S_{n}$ of an arithmetic series.

Worked examples

Easy: $a_{1} = 7$, $d = -3$, find $a_{10}$

From the first term to the tenth there are 9 steps of $-3$:

Medium: $a_{1} = 3$, $r = 2$, find $a_{6}$

From the first term to the sixth there are 5 multiplications by 2:

Hard: $a_{1} = 5$, $d = 4$, find $S_{10}$

Another way: the last term is $a_{10} = 5 + 9 \cdot 4 = 41$, and the sum is the number of terms times the average of the first and last term, $10 \cdot \tfrac{5 + 41}{2} = 230$.

Check your answers

For small cases, write the terms out. The geometric example is 3, 6, 12, 24, 48, 96, and the sixth term is 96.

Common mistakes

• Using $n$ instead of $n - 1$. $7 + 10(-3) = -23$ is the eleventh term, not the tenth.
• The same slip in geometric sequences. $3 \cdot 2^{6} = 192$ is $a_{7}$, not $a_{6}$.
• Signs with a negative ratio. With $a_{1} = 1$ and $r = -2$, the terms alternate: 1, −2, 4, −8. So $a_{4} = (-2)^{3} = -8$.
• Answering with a term when a sum is asked. $a_{10} = 41$ but $S_{10} = 230$.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

1. Arithmetic: a₁ = 3, d = 8
   Find a₁₀
2. Arithmetic: a₁ = 1, d = 5
   Find a₁₄
3. Arithmetic: a₁ = 3, d = 8
   Find a₁₂

Medium

4. Geometric: a₁ = 4, r = -2
   Find a₅
5. Geometric: a₁ = 2, r = -2
   Find a₆
6. Geometric: a₁ = 1, r = 3
   Find a₆

Hard

7. Arithmetic series: a₁ = 8, d = 2
   Find S₁₃
8. Arithmetic series: a₁ = 7, d = 2
   Find S₁₁
9. Arithmetic series: a₁ = 3, d = 5
   Find S₁₄

Tips for teachers and parents

• Before using a formula, ask students to write the first four terms. It makes the $n - 1$ in the formula obvious.
• The story of young Gauss adding 1 to 100 by pairing numbers is a good introduction to the series formula on Hard.
• Comparing an arithmetic and a geometric sequence with the same first term shows how quickly repeated multiplication overtakes repeated addition.

Related topics

• Functions Pre-Calculus
• Exponential Equations Algebra 2
• Solving Equations Algebra