Sequences and Series

Toolkit

Arithmetic Sequences

Arithmetic sequences are sequences with a common difference, that is to say, that the difference between consecutive terms is constant.

Description	Formula
General $k^{t h}$ term	$t_{k} = a + (k - 1) d$ , where $a$ is the first term and $d$ is the common difference
Sum of $n$ terms	$S_{n} = \frac{n}{2} (a + t_{n}) = \frac{n}{2} (2 a + (n - 1) d)$
Spacing of terms	Because there is a common difference between consecutive terms we have $t_{k} + t_{l} = t_{m} + t_{n}$ if and only if $k + l = m + n$

Geometric Sequences

Geometric sequences are sequences with a common ratio, that is to say, the ratio of consecutive terms is constant.

Description	Formula
General $k^{t h}$ term	$t_{k} = a r^{k - 1}$ , where $a$ is the first term and $r$ is the common ratio
Sum of $n$ terms	$S_{n} = \frac{a (1 - r^{n})}{(1 - r)}$
Spacing of terms	If $a \neq 0$ and $r \neq 0$ , then because there is a common ratio between consecutive terms, we have $t_{k} t_{l} = t_{m} t_{n}$ if and only if $k + l = m + n$ .
Infinite sum	If the ratio $r$ satisfies the condition $\| r \| < 1$ , we can calculate the infinite sum $a + a r + a r^{2} + a r^{3} + \dots$ using $S = \frac{a}{1 - r}$ .

Other

Arithmetic and geometric sequences are a small subset of all sequences, even though they are emphasized in high school mathematics. The following are some extensions that frequently appear on contests.

Description	Formula
Sum of the first $n$ integers	$\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}$
Sum of the first $n$ squares	$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$
Sum of the first $n$ cubes	$\sum_{k = 1}^{n} k^{3} = (\frac{n (n + 1)}{2})^{2}$
Telescoping series	If $t_{k} = u_{k} - u_{k - 1}$ , then $\sum_{k = 1}^{n} t_{k} = \sum_{k = 1}^{n} (u_{k} - u_{k - 1}) = u_{n} - u_{0}$

A recursive sequence is a sequence in which each term can be defined in relation to the previous term (or multiple previous terms).
- A recursion formula defines how to calculate each term $t_{n}$ from the previous term (or multiple previous terms).
- Every recursion formula needs to include at least one known or given term. This term is often the initial term, or $t_{1}$ .
- The general term of a sequence is an expression that is used to calculate each term, $t_{n}$ , in a sequence directly from its term number, $n$ . This representation of the sequence is sometimes called the closed form.

Sample Problems

What is the sum of all multiples of $7$ or $11$ less than $1000$ ?

Solution

Since we are adding $(7 + 14 + 21 + 28 + \dots + 994) + (11 + 22 + 33 + \dots + 990)$ , we are adding two arithmetic sequences. However the multiples of $77$ are included in both sequences and so must be subtracted (in order to avoid counting them twice) after we add the two sequences above. Therefore, the required sum is $(7 + 14 + 21 + 28 + \dots + 994) + (11 + 22 + 33 + \dots + 990) - (77 + 154 + \dots + 924) .$

The term $994$ is the 142nd term in the first sequence and so the sum of the first sequence is $\frac{142}{2} (7 + 994)$ . The term $990$ is the 90th term in the second sequence and so the sum of the second sequence is $\frac{90}{2} (11 + 990)$ . The term $924$ is the 12th term in the sequence of terms we remove and so the sum of that sequence is $\frac{12}{2} (77 + 924)$ . Thus, the required sum is $\begin{aligned} \frac{142}{2} (7 + 994) + \frac{90}{2} (11 + 990) - \frac{12}{2} (77 + 924) \\ = (71 + 45 - 6) (1001) \\ = (110) (1001) \\ = 110110 \end{aligned}$
A sequence is given such that $t_{1} = 1$ and $t_{n + 1} = t_{n} + 3 n^{2} + 3 n + 1$ . Evaluate $t_{100}$ .

Solution

Since the difference, $t_{n} - t_{n - 1}$ is not constant, the series is not arithmetic. Setting $n = 1$ , we find $t_{2} = 1 + 3 + 3 + 1 = 8$ Setting $n = 2$ , we find $t_{3} = 8 + 12 + 6 + 1 = 27$ These facts suggest $t_{n} = n^{3}$ for every $n$ .

To prove that $t_{n} = n^{3}$ is an alternate definition for the same sequence, we first note that $t_{1} = 1 = 1^{3}$ . Further, consider two adjacent terms in the sequence given by the alternate definition, i.e. $t_{n} = n^{3}$ and $t_{n + 1} = (n + 1)^{3}$ . Then the difference between these terms is $\begin{aligned} t_{n + 1} - t_{n} & = (n + 1)^{3} - (n)^{3} \\ = (n^{3} + 3 n^{2} + 3 n + 1) - n^{3} \\ = 3 n^{2} + 3 n + 1 \\ t_{n + 1} & = t_{n} + 3 n^{2} + 3 n + 1 \end{aligned}$ which matches the original definition of the sequence. We have proved that the original sequence can be expressed as $t_{n} = n^{3}$ , and thus, $t_{100} = 100^{3}$ .
If $a$ , $b$ , $a + b$ , and $a b$ are positive numbers that form $4$ consecutive terms in a geometric sequence, find $a$ .
Solution

Since we have a geometric sequence, the ratios of consecutive terms will be equal. So $\begin{aligned} (*) & \frac{a}{b} & = \frac{b}{a + b} = \frac{a + b}{a b} \end{aligned}$ Therefore, $\begin{aligned} \frac{a}{b} & = \frac{b}{a + b} \\ a^{2} + a b & = b^{2} \\ b^{2} - a b - a^{2} & = 0 \\ {(\frac{b}{a})}^{2} - (\frac{b}{a}) - 1 & = 0 & (since a \neq 0) \\ \frac{b}{a} & = \frac{1 + \sqrt{5}}{2} \end{aligned}$ where we have chosen the positive root since $a$ and $b$ are positive. Also from ( $*$ ), $\begin{aligned} \frac{a}{b} & = \frac{a + b}{a b} \\ a^{2} & = a + b \\ a & = 1 + \frac{b}{a} & (since a \neq 0) \\ = \frac{3 + \sqrt{5}}{2} & (substituting from above) \end{aligned}$

Problem Set

In a geometric series, $t_{5} + t_{7} = 1500$ and $t_{11} + t_{13} = 187500$ . Find all possible values for the first three terms.
Given that $a$ , $b$ and $c$ are consecutive terms in an arithmetic sequence that has distinct terms, calculate $x$ if $(b - c) x^{2} + (c - a) x + (a - b) = 0$
If $x$ , $4$ , $y$ are consecutive terms in an arithmetic sequence and $x$ , $3$ , $y$ are consecutive terms in a geometric sequence, calculate $\frac{1}{x} + \frac{1}{y}$ .
Three different numbers, whose product is $125$ , are $3$ consecutive terms in a geometric sequence. At the same time they are the first, third and sixth terms of an arithmetic sequence. Find these three numbers.
The $k$ th triangular number is given by $T_{k} = 1 + 2 + 3 + \dots + k = \frac{k (k + 1)}{2} = \frac{k^{2} + k}{2}$ . The first six triangular numbers are $1$ , $3$ , $6$ , $10$ , $15$ , and $21$ . Find the sum of the first $200$ triangular numbers.
If the interior angles of a pentagon form an arithmetic sequence and one interior angle is $90 °$ , find all possible values of the largest angle in the pentagon.
Find the four integers $a$ , $b$ , $c$ and $d$ that satisfy the following conditions:
- the sum of $b$ and $c$ is $30$
- the sum of $a$ and $d$ is $35$
- the numbers $a < b < c < d$ are in geometric sequence
A sequence $t_{1}$ , $t_{2}$ , $t_{3}$ is formed by choosing $t_{1}$ at random from the set ${1, 2, 3}$ , $t_{2}$ at random from the set ${4, 5, 6}$ , and $t_{3}$ at random from the set ${7, 8, 9}$ . What is the probability that $t_{1}$ , $t_{2}$ , $t_{3}$ is an arithmetic sequence?
The sum of $25$ consecutive integers is $500$ . Determine the smallest of the $25$ integers.
What is the number of terms in the arithmetic sequence $- 1994, - 1992, - 1990, \dots, 1992, 1994$ ?
The sum of the first $n$ terms of a sequence is $S_{n} = 3^{n} - 1$ , where $n$ is a positive integer.
1. If $t_{n}$ represents the $n$ th term of the sequence, determine $t_{1}$ , $t_{2}$ , $t_{3}$ .
2. Prove that $\frac{t_{n + 1}}{t_{n}}$ is constant for all values of $n$ .
How many terms in the arithmetic sequence $7, 14, 21, \dots$ are between $40$ and $28 001$ ?
If $f$ is a function such that $f (1) = 2$ and $f (n + 1) = \frac{3 f (n) + 1}{3}$ for $n = 1$ , 2, 3, $\dots$ , what is the value of $f (100)$ ?
Consider the family of lines with equations of the form $p x + q y = r$ , and which all pass through the point $(- 1, 2)$ . Prove that $p$ , $q$ , and $r$ are consecutive terms of an arithmetic sequence.
An arithmetic sequence S has terms $t_{1}, t_{2}, t_{3}, \dots$ , where $t_{1} = a$ and the common difference is d. The terms $t_{5}$ , $t_{9}$ , and $t_{16}$ form a three-term geometric sequence with common ratio r. Prove that S contains an infinite number of three-term geometric sequences, all having the same common ratio r.
In the sequence $5, 3, - 2, - 5, \dots$ , each term after the first two is constructed by taking the preceding term and subtracting the term before it. What is the sum of the first $32$ terms in the sequence?
Consider the sequence $t_{1} = 1$ , $t_{2} = - 1$ and $t_{n} = (\frac{n - 3}{n - 1}) t_{n - 2}$ where $n \geq 3$ . What is the value of $t_{1998}$ ?
The $n$ th term of an arithmetic sequence is given by $t_{n} = 555 - 7 n$ . If $S_{n} = t_{1} + t_{2} + \dots + t_{n}$ , determine the smallest value of $n$ for which $S_{n} < 0$ .