Sequences and Series

Precalculus is the course that gets functions ready for calculus. A function is a rule with an input and one output, and here you learn how to graph, change, and reverse them.

Sequences and Series: the central idea

A sequence is an ordered list generated explicitly or recursively. A series is the sum of sequence terms, so its convergence asks a different question.

Words you need

Sequence

A sequence is an ordered list of numbers where each position has an index, so $a_1$ is the first term, $a_2$ is the second, and so on.

Series

A series is the sum of the terms of a sequence, so the sequence $5,15,45$ has the matching series $5+15+45=65$.

Common difference

The common difference $d$ is the fixed number added to move from one term to the next in an arithmetic sequence, found by subtracting any term from the one after it.

Common ratio

The common ratio $r$ is the fixed number each term is multiplied by in a geometric sequence, found by dividing any term by the one before it.

Recursive rule

A recursive rule defines a term using the term before it, such as $a_1=5$ and $a_n=3a_{n-1}$, so you must walk down the list rather than jump straight to term $50$.

Convergence

Convergence means the partial sums of an infinite series settle down toward one fixed number instead of growing without bound or bouncing around forever.

What to know before this lesson

Know function notation, exponent rules, finite sums, and the difference between a term and an accumulated total.

If one of those prerequisites is uncertain, use the Precalculus subject guide to locate the earlier concept before memorizing a procedure.

Sequences and series: a worked example

Every step, with the arithmetic

1. Step 1 - Look at the list$5,\;15,\;45,\;135,\;\ldots$
2. Step 2 - Test for a common ratio$15\div5=3$ and $45\div15=3$ and $135\div45=3$, so $r=3$ and the sequence is geometric
3. Step 3 - Write the term rule$a_n=5\cdot3^{\,n-1}$, with the exponent $n-1$ because the first term has been multiplied zero times
4. Step 4 - Find the sixth term$a_6=5\cdot3^5=5\cdot243=1215$
5. Step 5 - Sum the first six with the formula$S_6=\dfrac{5(3^6-1)}{3-1}=\dfrac{5(729-1)}{2}=\dfrac{3640}{2}=1820$
6. Step 6 - Add them by hand to confirm$5+15+45+135+405+1215=1820$
7. Step 7 - Ask about the infinite sum$|r|=3$ is not less than $1$, so the terms grow and the infinite series diverges
8. Step 8 - Change the ratio to $1/3$ and retry$5+\tfrac53+\tfrac59+\cdots=\dfrac{5}{1-\tfrac13}=\dfrac{5}{2/3}=7.5$

With $a_1=3$ and common ratio $2$, multiplying by $2$ at each step gives $a_n=3\cdot2^{n-1}$; the exponent is $n-1$ because the first term has zero multiplications.

Term rules and sum formulas for the sequences that come up most

The step-by-step method for sequences and series

1. Identify whether successive terms have a constant difference, constant ratio, or another recurrence.
2. Write the term rule with correct indexing and evaluate the requested term before summing.
3. For an infinite series, test convergence before using a sum formula.

How to check your answer

Generate the first four terms from the formula and compare their differences or ratios with the stated pattern.

Use intercepts, end behavior, symmetry, and a few exact points to test whether a formula and graph describe the same function.

A mistake that changes the mathematics

A sequence can approach zero while its series diverges; $a_n\to0$ is necessary but not sufficient for $\sum a_n$ to converge.

Where you will actually use this

Loan and savings schedules

Each monthly payment is a term of a geometric series, which is how a bank computes that borrowing $10{,}000$ at a fixed rate costs a specific total over $60$ months.

Medicine in the bloodstream

If a body clears $25\%$ of a dose each day, the leftover amounts form a geometric sequence with $r=0.75$, and the infinite sum gives the long-run steady level from repeated dosing.

Bouncing ball and shrinking distances

A ball dropped from $2$ metres that rebounds to $60\%$ of its height travels a finite total distance, because the sum $2+2(0.6)+2(0.36)+\cdots$ converges.

How sequences and series connects to the rest of precalculus

• Exponentials and logarithms — A geometric sequence is an exponential function sampled at whole-number inputs, so $a_n=5\cdot3^{n-1}$ and $y=5\cdot3^{x-1}$ are the same rule.
• Understanding limits — Saying a series converges is a statement about the limit of its partial sums, so limits are the tool that makes convergence precise.
• Ratios and proportions — Spotting a common ratio in a sequence is the same skill as checking whether two ratios describe the same multiplicative jump.

Try a transfer problem

Compare the sequence $1/n$ with the series $\sum1/n$, describing what happens to terms and partial sums separately.

Show the worked answer

The sequence $a_n=1/n$ and the series $\sum 1/n$ behave in opposite ways, and comparing them side by side is the point. The terms shrink: $a_1=1$, $a_{10}=0.1$, $a_{100}=0.01$, $a_{1000}=0.001$. They head to $0$, so the sequence converges to $0$. The partial sums do not shrink at all: $H_1=1$, $H_{10}\approx2.929$, $H_{100}\approx5.187$, $H_{1000}\approx7.485$, $H_{1{,}000{,}000}\approx14.39$. They keep climbing, just slower and slower, and they pass every number you name eventually. This is the harmonic series, and it diverges. The reason is grouping: $\tfrac13+\tfrac14$ is more than $\tfrac12$, and $\tfrac15+\tfrac16+\tfrac17+\tfrac18$ is also more than $\tfrac12$. You can keep assembling blocks worth at least $\tfrac12$ forever, so the total has no ceiling. The lesson: terms going to zero is necessary for a series to converge, but it is not enough. $1/n$ shrinks, yet too slowly, while $1/2^n$ shrinks fast enough to sum to exactly $1$.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to precalculus practice questions.

Questions about sequences and series

How do I tell an arithmetic sequence from a geometric one?

Subtract neighbouring terms and also divide them. For $4,7,10,13$ the differences are all $3$, so it is arithmetic. For $4,8,16,32$ the ratios are all $2$, so it is geometric. If neither test gives a constant, it is some other pattern, such as $1,4,9,16$, which is the square numbers.

Why does the exponent say $n-1$ instead of $n$?

Because the first term has not been multiplied by the ratio yet. With $a_1=5$ and $r=3$, term $1$ is $5\cdot3^0=5$ and term $4$ is $5\cdot3^3=135$, which is three multiplications after the start. Using $3^n$ would shift the whole list one place and make $a_1=15$.

When can I use the infinite sum formula $a_1/(1-r)$?

Only when $|r|<1$, so the terms keep shrinking. With $r=0.5$ and $a_1=8$ the sum is $8/0.5=16$. With $r=2$ the formula would report $8/(-1)=-8$, which is nonsense for a list of growing positive numbers, and the real answer is that the series diverges.

What is sigma notation actually telling me?

$\sum_{n=1}^{5}2n$ is an instruction: plug in $n=1,2,3,4,5$, then add the results. That gives $2+4+6+8+10=30$. The letter under the sigma is the counter, the number below is where it starts, and the number on top is where it stops.