10 Sequences and Series

Core Ideas: Sequences and Series

A Sequence is an ordered list of numbers, while a Series is formed by adding sequence terms. The terms are labeled by position: the term in position $n$ is written as $a_n$. For example, the list $2,5,8,11,\ldots$ has $a_1=2$, $a_2=5$, and $a_3=8$.

A sequence can be described in several ways:

• A list of terms shows the pattern directly.

• A table or graph connects positions with values.

• A Recursive formula explains how to generate each term from earlier terms.

• An Explicit formula gives a term directly from its position.

A series may be written in expanded form, such as $a_1+a_2+a_3$, or compactly with summation notation.

Takeaway: A sequence describes values in order; a series describes their sum.

Two Ways to Describe a Sequence

A recursive description requires an initial value and a rule for producing later terms. For the sequence $4,7,10,13,\ldots$, one recursive representation is

The terms are generated in order:

An explicit description gives the same sequence directly:

Thus, a distant term can be found without calculating all the previous terms:

Use a recursive formula when the generation process matters or when terms are built sequentially. Use an explicit formula when a particular, especially distant, term is needed efficiently.

Takeaway: Recursive formulas emphasize construction; explicit formulas emphasize direct calculation.

Arithmetic Patterns

An Arithmetic sequence has a constant additive change between consecutive terms. This constant is the Common difference, denoted by $d$:

For example, $12,17,22,27,\ldots$ is arithmetic because each term increases by $5$. Its recursive form is

Its explicit form is

The factor $n-1$ counts the changes made after the first term. For $7,11,15,19,\ldots$, the first term is $7$ and the common difference is $4$, so

If two terms are known, the common difference can be calculated using

Takeaway: Constant subtraction of consecutive terms identifies an arithmetic sequence, and its terms follow $a_n=a_1+(n-1)d$.

Geometric Patterns

A Geometric sequence has a constant multiplicative change between consecutive terms. This factor is the Common ratio, denoted by $r$:

For example, $3,12,48,192,\ldots$ is geometric because each term is multiplied by $4$. Its recursive form is

and its explicit form is

For $5,10,20,40,\ldots$, the first term is $5$ and the common ratio is $2$. Therefore,

The value of the ratio determines the pattern:

• If $\lvert r\rvert>1$, the magnitudes generally grow.

• If $0<\lvert r\rvert<1$, the magnitudes generally decay.

• If $r<0$, the signs alternate.

Takeaway: Constant division of consecutive terms identifies a geometric sequence, and its terms follow $a_n=a_1r^{n-1}$.

Compact Addition with Sigma Notation

Summation notation uses the Greek letter sigma to express a long addition compactly. In

$k$ is the index of summation, $1$ is the starting value, $n$ is the ending value, and $a_k$ is the general term.

For example,

Evaluating the terms gives

An arithmetic list can also be represented with summation notation:

Takeaway: Read a sigma expression by identifying its index, starting value, ending value, and general term.

Adding Finite Patterns

An Arithmetic series adds the first $n$ terms of an arithmetic sequence. Its main sum formula is

When the common difference is known, use the equivalent form

These formulas work because terms can be paired from opposite ends, and each pair has the same sum.

For the sequence $6,10,14,18,\ldots$, the first term is $6$, the common difference is $4$, and the twentieth term is

Therefore, the sum of the first twenty terms is

A finite geometric series adds a fixed number of terms of a geometric sequence. For $r\ne1$, use

For the first eight terms of $2,6,18,54,\ldots$, where $a_1=2$ and $r=3$,

If $r=1$, every term equals $a_1$, so $S_n=na_1$.

Takeaway: Use the arithmetic sum formula for constant differences and the finite geometric sum formula for constant ratios.

Infinite Geometric Sums

An Infinite geometric series has a finite sum only when the common ratio satisfies

In that case,

For example, the series

has $a_1=12$ and $r=\frac{1}{2}$, so

When $\lvert r\rvert\ge1$, the terms do not approach zero in the required way, so the infinite series does not converge to a finite sum.

Takeaway: Before applying the infinite-sum formula, always check that $\lvert r\rvert<1$.

Selecting and Applying a Model

Choose a model by examining how the quantity changes from one step to the next.

1. Subtract consecutive values. A constant difference suggests an arithmetic model.

2. Divide consecutive nonzero values. A constant ratio suggests a geometric model.

3. Check the context. Fixed additions or subtractions suggest arithmetic change, while repeated percentage changes suggest geometric change.

4. Use the selected formula to calculate or predict values, remembering that real data may only approximately follow a pattern.

For a constant-addition situation, such as weekly deposits beginning at $40$ dollars and increasing by $5$ dollars each week,

The twelfth deposit is

and the total deposited over twelve weeks is

For a repeated percentage decrease, convert the percentage to a multiplier. A decrease of $12$ percent means multiplying by $0.88$. If the initial value is $18{,}000$, the value after $n$ years is modeled by

Takeaway: Constant additive change calls for an arithmetic model; constant multiplicative or percentage change calls for a geometric model.