Arithmetic Sequences and Series Worked Examples Cheat Sheet

Arithmetic sequences and series describe patterns that change by adding the same amount each time. This cheat sheet helps students identify the common difference, write formulas, find specific terms, and calculate sums. It is useful for homework, test review, and worked-example practice because the same few formulas appear in many problems.

The main ideas are the first term $a_1$ , the common difference $d$ , the term number $n$ , and the partial sum $S_n$ . An arithmetic sequence uses $a_n = a_1 + (n - 1)d$ to find any term. An arithmetic series uses $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n - 1)d)$ to add the first $n$ terms.

Key Facts

An arithmetic sequence has a constant common difference $d$ , so $d = a_n - a_{n-1}$ for consecutive terms.
The explicit formula for the $n$ th term is $a_n = a_1 + (n - 1)d$ .
The recursive formula is $a_1$ given and $a_n = a_{n-1} + d$ for $n \ge 2$ .
To find the term number, rearrange $a_n = a_1 + (n - 1)d$ and solve for $n$ .
The sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$ when the first and last terms are known.
The sum formula $S_n = \frac{n}{2}(2a_1 + (n - 1)d)$ is useful when $a_1$ , $d$ , and $n$ are known.
For the sequence $7, 11, 15, 19, \ldots$ , the common difference is $d = 4$ and the explicit formula is $a_n = 7 + 4(n - 1)$ .
For an arithmetic series, the average of the first and last terms is $\frac{a_1 + a_n}{2}$ , so $S_n = n \cdot \frac{a_1 + a_n}{2}$ .

Vocabulary

Arithmetic sequence: An arithmetic sequence is a list of numbers where each term is found by adding the same common difference $d$ .
Common difference: The common difference $d$ is the constant amount added to one term to get the next term.
Explicit formula: An explicit formula gives the value of $a_n$ directly using the term number $n$ .
Recursive formula: A recursive formula defines each term using the previous term, such as $a_n = a_{n-1} + d$ .
Arithmetic series: An arithmetic series is the sum of the terms of an arithmetic sequence.
Partial sum: A partial sum $S_n$ is the sum of the first $n$ terms of a sequence.

Common Mistakes to Avoid

Using $n$ instead of $n - 1$ in the explicit formula is wrong because the first term already occurs when $n = 1$ .
Finding $d$ by subtracting nonconsecutive terms without dividing by the number of steps is wrong because $d$ is the change per term.
Using the series formula for one term is wrong because $S_n$ means the sum of terms, not the value of $a_n$ .
Forgetting that a decreasing arithmetic sequence has a negative common difference is wrong because $d$ must match the direction of the pattern.
Substituting the last term for $n$ is wrong because $a_n$ is a term value while $n$ is the term number.

Practice Questions

1 Find $a_{20}$ for the arithmetic sequence with $a_1 = 6$ and $d = 3$ .
2 Find the sum $S_{15}$ for the arithmetic series with $a_1 = 4$ and $d = 5$ .
3 The sequence $42, 37, 32, 27, \ldots$ is arithmetic. Write an explicit formula for $a_n$ and find $a_{12}$ .
4 Explain how you can tell whether a sequence is arithmetic, and describe why the same test works for increasing and decreasing sequences.

Understanding Arithmetic Sequences and Series Worked Examples

A sequence is a list of individual values. A series is the total produced when selected values from that list are added. Keeping those ideas separate prevents a common mistake.

If a problem asks for the twentieth value, the answer is one number from the pattern. If it asks for the total through the twentieth value, the answer combines twenty values. Arithmetic patterns create straight-line graphs when term position is placed on the horizontal axis and term value is placed on the vertical axis.

The common difference becomes the slope. A positive difference makes the graph rise.

A negative difference makes it fall. A difference of zero gives a flat line.

The sum rule comes from pairing terms at opposite ends of a finite list. In the list two, five, eight, eleven, fourteen, the first and last make sixteen. The second and second-to-last make sixteen too.

Each pair has the same total because one term moves upward by the same amount that the other moves downward. This is why the average of the first and last values is so useful. The total equals the number of terms times that average.

When there is an odd number of terms, the middle term has no partner, but it fits the same average idea. Understanding this pairing method is better than memorising a rule with no reason behind it.

Arithmetic sequences appear whenever a quantity changes by a fixed amount for each equal step. A savings plan might gain the same number of pounds each week. Seats in theatre rows may increase by a fixed number from row to row.

A taxi fare can start with a fixed charge, then rise by the same amount per kilometre under a simplified pricing model. In each setting, identify what one term represents before calculating. It could represent a weekly balance, a row size, or a cost.

Units matter. A difference of three seats per row is not the same kind of quantity as a difference of three pounds per week. Clear units help reveal whether an answer makes sense.

Worked problems become easier when the given information is organised before any calculation. Write down the starting value, the fixed change, and the position or number of terms. Check the direction of the pattern from at least two consecutive values instead of assuming it rises.

Be careful with counting. The first term is at position one, so reaching position ten means making nine changes from the start. This off-by-one detail causes many wrong answers.

When solving for a position, check whether the result is a positive whole number. A value can lie on the straight-line pattern without actually appearing as a listed term if its position is not a whole number.

Finally, substitute your answer back into the pattern or estimate its size. A term far from the start should reflect many repeated changes, while a sum of positive increasing terms should be larger than the final term.

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