Summary of Sequence Formulas

Sequence formulas are expressions used to quickly find the $n$th term or the sum of the first $n$ terms in a sequence that follows a specific rule. Since exams typically focus on arithmetic and geometric sequences, your first step should always be to determine whether the "difference is constant" or the "ratio is constant."

Here are the four core formulas you can use right away.

Sequence Formulas at a Glance

For an arithmetic sequence with a common difference of $d$:

$a_n = a_1 + (n-1)d$

$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}\{2a_1 + (n-1)d\}$

For a geometric sequence with a common ratio of $r$:

$a_n = a_1 r^{n-1}$

$S_n = \frac{a_1(1-r^n)}{1-r} \quad \text{if } r \ne 1$

And if $r = 1$, every term is $a_1$, so:

$S_n = na_1$

In these formulas, $a_n$ represents the $n$th term, and $S_n$ is the sum from the first term up to the $n$th term. Make sure to check exactly what the problem is asking for so you don't mix up the formulas.

How to Distinguish Between Arithmetic and Geometric Sequences

An arithmetic sequence is one where the difference between any two consecutive terms is constant. For example, in $3, 7, 11, 15, \dots$, the value increases by $4$ every time, so the common difference is $d=4$.

A geometric sequence is one where the ratio between any two consecutive terms is constant. For example, in $2, 6, 18, 54, \dots$, each term is $3$ times the previous one, so the common ratio is $r=3$.

This distinction comes first. If you use a geometric formula for a sequence with a constant difference, or an arithmetic formula for one with a constant ratio, all your subsequent calculations will be wrong.

Intuitive Look: Why the Formulas Look This Way

The general term of an arithmetic sequence, $a_n = a_1 + (n-1)d$, is the result of adding the common difference $n-1$ times to the first term. You can think of it as adding the same amount every time you move one step forward.

The general term of a geometric sequence, $a_n = a_1r^{n-1}$, is the result of multiplying the first term by the common ratio $n-1$ times. It's a structure where the value grows or shrinks by the same scale at each step.

Therefore, arithmetic sequences change by a constant amount, while geometric sequences involve cumulative change. However, in a geometric sequence, if the common ratio is $0 < r < 1$, the terms will gradually get smaller.

Example: Finding the General Term and the Sum

Let's look at the sequence $5, 8, 11, 14, \dots$.

Since the difference is always $3$, it is an arithmetic sequence with the first term $a_1=5$ and a common difference of $d=3$.

Finding the 10th Term

Using the general term formula for an arithmetic sequence:

$a_n = a_1 + (n-1)d$

Therefore:

$a_{10} = 5 + (10-1) \cdot 3 = 5 + 27 = 32$

So, the 10th term is $32$.

Finding the Sum of the First 10 Terms

Using the sum formula:

$S_n = \frac{n}{2}(a_1+a_n)$

And plugging in $n=10$, $a_1=5$, and $a_{10}=32$:

$S_{10} = \frac{10}{2}(5+32) = 5 \cdot 37 = 185$

The key in this example is distinguishing the symbols. $a_{10}$ is a single term, while $S_{10}$ is the total sum of the first 10 terms. Always read carefully to see if the problem asks for the "10th term" or the "sum of the first 10 terms."

Common Mistakes in Sequence Problems

Mixing Up the General Term and Sum Formulas

It's common to use the $S_n$ formula when you need to find $a_n$, or conversely, to calculate only the general term and stop when you were supposed to find the sum. Always verify if the question asks for "which term" or "the sum of how many terms."

Treating a Geometric Sequence as Arithmetic

For example, $2, 4, 8, 16, \dots$ is not an arithmetic sequence because the difference is not constant. For sequences like this, you must check the ratio.

Forgetting the Condition for the Geometric Sum Formula

$S_n = \frac{a_1(1-r^n)}{1-r}$

can only be used directly when $r \ne 1$. If $r=1$, the denominator becomes $0$, so it must be handled separately.

Forgetting $n-1$

In the general term formula, $a_n$ must represent the state where you haven't moved from the first term yet, which is $n=1$. That's why both arithmetic and geometric sequences include $n-1$.

When to Use Sequence Formulas

Sequence formulas are used not only in school exams but also when describing situations with constant growth or repeating ratios. For instance, a monthly savings plan that increases by a fixed amount is similar to an arithmetic sequence, and phenomena that increase or decrease by a fixed percentage can be modeled as a geometric sequence.

However, you must check the conditions to see if the real-world situation strictly follows an arithmetic or geometric pattern. Formulas can only be applied directly when the rule fits perfectly.

Step-by-Step Solving Process

You can usually solve sequence problems by following these steps:

1. Check if the difference is constant or if the ratio is constant.
2. Determine if the value you need to find is $a_n$ or $S_n$.
3. Find $d$ if it's an arithmetic sequence, or $r$ if it's a geometric sequence.
4. Plug the values into the appropriate formula.

Recommended Practice

Try finding $a_{20}$ and $S_{20}$ for the example sequence $5, 8, 11, 14, \dots$. Then, apply the same questions to the geometric sequence $3, 6, 12, 24, \dots$. This will help make it clearer when to use the general term formula versus the sum formula.