Algebra · Chapter 14

Sequences and Series

Spot the pattern, jump straight to the 100th term, and add up long lists in one step with Gauss's pairing trick.

What's this chapter about?

14.1 Arithmetic Sequences
14.2 Special Series
14.3 Geometric Sequences
14.4 Arithmetico-Geometric Sequence
14.5 Practice Problems

You'll need

14.10%

Arithmetic Sequences

In this section you will be able to find any term of an arithmetic sequence, count its terms, and add them all up fast.

Count 1, 4, 7, 10, 13, … What comes after 40? You add 3 every time. What would the 100th number be? You don’t want to count all the way there!

Each term is the one before it plus the same number $d$ . So to get from the first term to the $n$ th term, you take $n - 1$ jumps of size $d$ .

Look: the terms of an arithmetic sequence are evenly spaced. So their average sits right in the middle. It equals the average of the first and last terms. Then the sum is just the average times the number of terms.

Exploreplay first, formulas later

Try this: Set the difference to 3 and press + on terms until the last term is 40. How many terms did you need?
Try this: Make the difference −3. The dots now walk to the left. Is it still an arithmetic sequence?
Try this: Pick term k = 8. Before you look, predict a₈ from a₁ and d. How many jumps of d did you add?
Try this: Turn on Show average and sum. Where does the average line cross the bars when there is an odd number of terms?

Name it

Definition 14.1.1 · Arithmetic Sequence

An arithmetic sequence is a sequence of numbers with the same difference between consecutive terms.

1, 4, 7, 10, 13, \ldots, 40

is an arithmetic sequence because there is always a difference of $3$ between consecutive terms.

Remark 14.1.2

An arithmetic sequence can also have a negative common difference. For example, in the arithmetic sequence

40, 37, 34, \ldots, 4, 1

the common difference is $-3$ .

Definition 14.1.3 · Arithmetic Sequence Notation

In general, the terms of an arithmetic sequence can be represented as:

a_1, a_2, a_3, a_4, \ldots, a_n

where

$d$ is the common difference between consecutive terms
$n$ is the number of terms

Theorem 14.1.4 · nth term in an Arithmetic Sequence

a_n = a_1 + (n-1)d

which basically means the $n$ th term of an arithmetic sequence is equal to

\text{first term} + (\text{number of terms} - 1)(\text{common difference})

We also have that

a_n = a_m + (n-m)d

which means

\text{the } n\text{th term} = m\text{th term} + (\text{number of terms} - m)(\text{common difference})

Theorem 14.1.5 · Number of Terms in an Arithmetic Sequence

n = \frac{a_n - a_1}{d} + 1

Essentially,

\text{Number of Terms} = \frac{\text{Last Term} - \text{First Term}}{\text{Common Difference}} + 1

Theorem 14.1.6 · Average of Terms in an Arithmetic Sequence

\text{Average of Terms} = \frac{a_1 + a_n}{2}

\text{Average of Terms} = \frac{a_1 + a_2 + \cdots + a_n}{n}

which essentially means

\text{Average of Terms} = \frac{\text{First Term} + \text{Last Term}}{2}

\text{Average of Terms} = \frac{\text{Sum of all Terms}}{\text{Number of Terms}}

If the number of terms is even, the average $x$ = the average of the middle 2 terms.

If the number of terms is odd, the average $x$ = the middle term.

Theorem 14.1.7 · Sum of all Terms in an Arithmetic Sequence

S_n = \frac{a_1 + a_n}{2} \times n

which essentially means

\text{Sum of All Terms} = \text{Average of Terms} \times \text{Number of Terms}

We can also substitute $a_n = a_1 + (n-1)d$ to get

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

Watch itpredict, then reveal

Example 14.1

AIME

Find the value of $a_2 + a_4 + a_6 + a_8 + \ldots + a_{98}$ if $a_1, a_2, a_3 \ldots$ is an arithmetic progression with common difference $1$ , and $a_1 + a_2 + a_3 + \ldots + a_{98} = 137$ .

1
Split the 98 terms into the odd-numbered ones ( $a_1, a_3, \ldots, a_{97}$ ) and the even-numbered ones ( $a_2, a_4, \ldots, a_{98}$ ). How many terms are in each group?

Predict first:
Your prediction

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Example 14.2

The sum of the first $5$ terms of an arithmetic sequence is $65$ and the sum of the first $10$ terms of the same sequence is $255$ . Find the sum of the first $15$ terms of the sequence.

1
The sum of 5 terms is the average times 5. The average of 5 terms is the middle term, $a_3$ . What is $a_3$ ?

Predict first:
Your prediction

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Try it

Problem 14.5.7(additional)

AHSME

What is the $100$ th number in the arithmetic sequence $1, 5, 9, 13, 17, 21, 25, \ldots$

Problem 14.5.9(additional)

AMC 8

The sum of $25$ consecutive even integers is $10{,}000$ . What is the largest of these $25$ consecutive integers?

Problem 14.5.8(additional)

MATHCOUNTS

The non-negative integers $a$ , $b$ , $c$ , $d$ , and $e$ form an arithmetic sequence. If their sum is $440$ , what is the largest possible value for $e$ ?

Contest cornerreal contest problems

Problem 14.5.1

AMC 8

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2, 5, 8, 11, 14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$ . Each row and each column in this $5 \times 5$ array is an arithmetic sequence with five terms. The square in the center is labelled $X$ as shown. What is the value of $X$ ?

Problem 14.5.3

AMC 10

Suppose that $\{a_n\}$ is an arithmetic sequence with

a_1 + a_2 + \cdots + a_{100} = 100 \text{ and } a_{101} + a_{102} + \cdots + a_{200} = 200.

What is the value of $a_2 - a_1$ ?

Problem 14.5.5

AMC 10

In the five-sided star shown, the letters $A$ , $B$ , $C$ , $D$ , and $E$ are replaced by the numbers $3$ , $5$ , $6$ , $7$ , and $9$ , although not necessarily in this order. The sums of the numbers at the ends of the line segments $AB$ , $BC$ , $CD$ , $DE$ , and $EA$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence?

Checkpoint for section 14.1

Get 2 of 3 right to master this section. One try each, no hints.

14.20%

Special Series

In this section you will be able to add 1 + 2 + … + n, the odd numbers, the even numbers, the squares and the cubes with one formula each.

A teacher once asked a class to add every number from 1 to 100. Young Carl Gauss had the answer in seconds. His trick: pair the first number with the last!

Look: $1 + 100$ , $2 + 99$ , $3 + 98$ … every pair adds to $101$ . Writing the sum twice, once forwards and once backwards, gives $n$ columns of $n + 1$ . Half of that is the sum.

Some sums show up again and again in contests. Learn these five and you can add a long list in one line.

Exploreplay first, formulas later

Try this: Press Fold. A flipped copy lands on top. How tall is every column? How many columns are there?
Try this: The picture holds two copies of the sum. So what is 1 + 2 + … + 10?
Try this: Set the first term to 1 and the difference to 2. Add the odd numbers for n = 3, 4, 5. What kind of numbers do you get?
Try this: Now start at 2 with difference 2. How does the sum of the even numbers compare with 1 + 2 + … + n?

Name it

Theorem 14.2.1 · Sum of Numbers Formula

1 + 2 + 3 + \cdots + n = \frac{(n)(n+1)}{2}

Theorem 14.2.2 · Sum of Odd Numbers Formula

1 + 3 + 5 + \cdots + (2n-1) = n^2

In simple terms, the

\text{Sum of first } n \text{ odd numbers} = n^2

Theorem 14.2.3 · Sum of Even Numbers Formula

2 + 4 + 6 + \cdots + 2n = n(n+1)

To intuitively think about it, just take $2$ common from each term:

2(1 + 2 + 3 + \cdots + n) = 2\,\frac{(n)(n+1)}{2} = n(n+1)

In simple terms, the

\text{Sum of first } n \text{ even numbers} = 2 \times \text{sum of first } n \text{ numbers}

Theorem 14.2.4 · Sum of Squares Formula

1^2 + 2^2 + \cdots + n^2 = \frac{(n)(n+1)(2n+1)}{6}

Theorem 14.2.5 · Sum of Cubes Formula

1^3 + 2^3 + \cdots + n^3 = \left(\frac{(n)(n+1)}{2}\right)^2

Try it

Problem 14.5.10(additional)

AMC 10/12

A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $100$ cans, how many rows does it contain?

Contest cornerreal contest problems

Problem 14.5.6

AIME

The terms of an arithmetic sequence add to $715$ . The first term of the sequence is increased by $1$ , the second term is increased by $3$ , the third term is increased by $5$ , and in general, the $k$ th term is increased by the $k$ th odd positive integer. The terms of the new sequence add to $836$ . Find the sum of the first, last, and middle terms of the original sequence.

Checkpoint for section 14.2

Get 2 of 3 right to master this section. One try each, no hints.

14.30%

Geometric Sequences

In this section you will be able to find terms and sums of geometric sequences, even sums that go on forever.

Fold a sheet of paper in half, then in half again. Each fold doubles the layers: 1, 2, 4, 8, 16. After 10 folds you have 1024 layers!

In a geometric sequence you multiply by the same number $r$ each time, instead of adding. To reach the $n$ th term you multiply by $r$ exactly $n - 1$ times.

When $r$ is between $-1$ and $1$ , the terms shrink so fast that the total stops growing. It creeps up on a fixed number: eat half a pizza, then half of what’s left, and so on. You never eat more than one pizza.

Exploreplay first, formulas later

Try this: With ratio 2, each bar doubles. How many terms does it take to reach 1024?
Try this: Change the ratio to −2. What happens to the signs of the terms?
Try this: Set the ratio to 1/2. Watch the running total. How big is the gap to the limit after each new term?
Try this: Try ratio 2 again with Show the limit on. Why is there no limit line now?

Name it

Definition 14.3.1 · Geometric Sequence

A geometric sequence is a sequence of numbers with the same ratio between consecutive terms.

1, 2, 4, 8, 16, 32, \ldots, 1024

is a geometric sequence because there is always a ratio of $2$ between consecutive terms.

Definition 14.3.2 · Geometric Sequence Notation

In general, the terms of a geometric sequence can be represented as:

g_1, g_2, g_3, g_4, \ldots, g_n

where

$r$ is the common ratio between consecutive terms
$n$ is the number of terms

Remark 14.3.3

A geometric sequence can also have a negative common ratio. For example the sequence $1, -2, 4, -8, \ldots, 512, -1024$ has a common ratio of $-2$ .

Theorem 14.3.4 · nth term in a Geometric Sequence

g_n = g_1 \cdot r^{n-1}

which basically means

\text{the } n\text{th term of a geometric sequence} = \text{first term} \times (\text{common ratio})^{\text{number of terms} - 1}

A general form for calculating the $n$ th term:

g_n = g_m \cdot r^{(n-m)}

which basically means

\text{the } n\text{th term} = m\text{th term} \times \text{common ratio}^{(n-m)}

Theorem 14.3.5 · Number of Terms in a Finite Geometric Sequence

n = \log_r\left(\frac{g_n}{g_1}\right) + 1

Essentially,

Number of Terms = $1$ more than the number of times we needed to multiply $r$ by $g_1$ to get $g_n$ .

Theorem 14.3.6 · Sum of all Terms in a Finite Geometric Sequence

S_n = g_1\,\frac{(1 - r^n)}{1 - r}

which essentially means

\text{Sum of All Terms} = \text{First Term} \times \frac{1 - \text{common ratio}^n}{1 - \text{common ratio}}

Theorem 14.3.7 · Sum of all Terms in an Infinite Geometric Sequence

For $-1 < r < 1$ ,

S_\infty = \frac{g_1}{1 - r}

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Example 14.3

AIME

Call a 3-digit number geometric if it has 3 distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

1
Write the digits as $a$ , $ar$ , $ar^2$ . For the largest number, what is the best first digit to try?

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Example 14.4

AIME

Suppose $n$ is a positive integer and $d$ is a single digit in base $10$ . Find $n$ if

\frac{n}{810} = 0.d25d25d25\ldots

1
The repeating block $\overline{d25}$ is the three-digit number $100d + 25$ . The decimal is $\frac{100d+25}{1000} + \frac{100d+25}{1000^2} + \cdots$ What is the common ratio of this geometric series?

Predict first:
Your prediction

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Contest cornerreal contest problems

Problem 14.5.2

Omega Learn Math Contest

There are $4$ positive integers $A$ , $B$ , $C$ , and $D$ . The first $3$ numbers form an arithmetic sequence in that order. The last $3$ numbers form a geometric sequence in that order. The value of $C$ is $3$ times the value of $A$ . What is the smallest possible value of $D$ ?

Problem 14.5.4

AMC 10/12

A sequence of three real numbers forms an arithmetic progression with a first term of $9$ . If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?

Checkpoint for section 14.3

Get 2 of 3 right to master this section. One try each, no hints.

14.40%

Arithmetico-Geometric Sequence

In this section you will be able to add series whose tops grow by adding and bottoms grow by multiplying, with the shift-and-subtract trick.

What is 1/2 + 2/4 + 3/8 + 4/16 + …? The tops climb by 1 while the bottoms double. It is neither arithmetic nor geometric. It is a bit of both!

Call the whole sum $S$ . Dividing $S$ by $2$ makes a copy whose terms line up one column to the right. Subtract the copy: the tops in each column now differ by exactly $1$ , and a plain geometric series is left.

Exploreplay first, formulas later

Try this: Press Divide S by 2. The new row is the same series, slid one place to the right. Compare the columns.
Try this: Press Subtract the rows. What is left in every column? What kind of series is that?
Try this: Set r to 3 and do it again. Can you predict S before you press Subtract?

Name it

Concept 14.4.1

The strategy for these problems is just to break the sequence into multiple geometric sequences. Another strategy is to let the expression be $S$ . Then, divide $S$ by some number such that when you subtract it from $S$ , most of the terms cancel. This strategy is faster, but harder to figure out.

Watch itpredict, then reveal

Example 14.5

Evaluate $\frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \frac{5}{32} + \ldots$

1
Call the sum $S$ . The tops go up by $1$ and the bottoms double. Halve every term of $S$ . What is the first term of $\frac{S}{2}$ ?

Predict first:
Your prediction

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Checkpoint for section 14.4

Get 2 of 3 right to master this section. One try each, no hints.

14.5

Practice Problems

In this section you will be able to mix arithmetic and geometric ideas on contest problems.

Every practice problem from this chapter already sits in the lesson it uses. Each one has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.