Arithmetic sequences are sequences with a common difference, that is to say, that the difference between consecutive terms is constant.
| Description | Formula |
|---|---|
| General \(k^{th}\) term | \(t_k=a+(k-1)d\), where \(a\) is the first term and \(d\) is the common difference |
| Sum of \(n\) terms | \(S_n=\dfrac{n}{2}(a+t_n)=\dfrac{n}{2}(2a+(n-1)d)\) |
| Spacing of terms | Because there is a common difference between consecutive terms we have \(\displaystyle t_k+t_l=t_m+t_n\) if and only if \(k+l=m+n\) |
Geometric sequences are sequences with a common ratio, that is to say, the ratio of consecutive terms is constant.
| Description | Formula |
|---|---|
| General \(k^{th}\) term | \(t_k=ar^{k-1}\), where \(a\) is the first term and \(r\) is the common ratio |
| Sum of \(n\) terms | \(S_n=\dfrac{a(1-r^n)}{(1-r)}\) |
| Spacing of terms | If \(a \neq 0\) and \(r \neq 0\), then because there is a common ratio between consecutive terms, we have \(t_kt_l=t_mt_n\) 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 \(a + ar + ar^2 + ar^3 + \cdots\) using \(S=\dfrac{a}{1-r}\). |
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 | \(\displaystyle\sum_{k=1}^{n}k=\dfrac{n(n+1)}{2}\) |
| Sum of the first \(n\) squares | \(\displaystyle\sum_{k=1}^{n}k^2=\dfrac{n(n+1)(2n+1)}{6}\) |
| Sum of the first \(n\) cubes | \(\displaystyle\sum_{k=1}^{n}k^3=\Biggr(\dfrac{n(n+1)}{2}\Biggr)^2\) |
| Telescoping series | If \(t_k=u_k-u_{k-1}\), then \(\displaystyle\sum_{k=1}^{n}t_k=\sum_{k=1}^{n}(u_k-u_{k-1})=u_n-u_0\) |
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.
What is the sum of all multiples of \(7\) or \(11\) less than \(1000\)?
Solution
Since we are adding \((7+14+21+28+ \cdots+994)+(11+22+33+\cdots+990)\), 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 \[(7+14+21+28+\cdots+994)+(11+22+33+\cdots+990)-(77+154+\cdots+924).\]
The term \(994\) is the 142nd term in the first sequence and so the sum of the first sequence is \(\dfrac{142}{2}(7+994)\). The term \(990\) is the 90th term in the second sequence and so the sum of the second sequence is \(\dfrac{90}{2}(11+990)\). The term \(924\) is the 12th term in the sequence of terms we remove and so the sum of that sequence is \(\dfrac{12}{2}(77+924)\). Thus, the required sum is \[\begin{align*} &\dfrac{142}{2}(7+994)+\dfrac{90}{2}(11+990)-\dfrac{12}{2}(77+924)\\ &=(71+45-6)(1001)\\ &=(110)(1001)\\ &=110110\end{align*}\]
A sequence is given such that \(t_1=1\) and \(t_{n+1}=t_n+3n^2+3n+1\). 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 \[\begin{align*} t_{n+1}-t_n&=(n+1)^3-(n)^3\\ &=(n^3+3n^2+3n+1)-n^3\\ &=3n^2+3n+1\\ t_{n+1}&=t_{n}+3n^2+3n+1\end{align*}\] 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 \(ab\) 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 \[\begin{align*} \dfrac{a}{b}&=\dfrac{b}{a+b}=\dfrac{a+b}{ab} \tag{$\ast$}\end{align*}\] Therefore, \[\begin{align*} \dfrac{a}{b}&=\dfrac{b}{a+b}\\ a^2+ab&=b^2\\ b^{2}-ab-a^{2}&=0\\ \left(\dfrac{b}{a}\right)^2-\left(\dfrac{b}{a}\right)-1&=0 & \text{(since $a \neq 0$)}\\ \dfrac{b}{a}&=\dfrac{1+\sqrt{5}}{2}\end{align*}\] where we have chosen the positive root since \(a\) and \(b\) are positive. Also from (\(\ast\)), \[\begin{align*} \dfrac{a}{b}&=\dfrac{a+b}{ab}\\ a^2&=a+b\\ a&=1+\dfrac{b}{a}& \text{(since $a \neq 0$)}\\ &=\dfrac{3+\sqrt{5}}{2}& \text{(substituting from above)}\\\end{align*}\]
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 \(\dfrac{1}{x}+\dfrac{1}{y}\).
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 \(T_{k}=1+2+3+ \cdots +k = \dfrac{k(k+1)}{2} = \dfrac{k^{2}+k}{2}\). 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 \(90\degree\), 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 \(-1994, -1992, -1990, \ldots, 1992, 1994\)?
The sum of the first \(n\) terms of a sequence is \(S_n=3^n-1\), where \(n\) is a positive integer.
If \(t_n\) represents the \(n\)th term of the sequence, determine \(t_1\), \(t_2\), \(t_3\).
Prove that \(\dfrac{t_{n+1}}{t_n}\) is constant for all values of \(n\).
How many terms in the arithmetic sequence \(7, 14, 21, \ldots\) are between \(40\) and \(28\,001\)?
If \(f\) is a function such that \(f(1)=2\) and \(f(n+1)=\dfrac{3f(n)+1}{3}\) for \(n=1\), 2, 3,\(\ldots\), what is the value of \(f(100)\)?
Consider the family of lines with equations of the form \(px+qy=r\), 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 \(t_1,t_2,t_3,\ldots\), 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 \(5,3,-2,-5, \ldots\), 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 \(t_n=\left(\dfrac{n-3}{n-1}\right)t_{n-2}\) where \(n\ge3\). What is the value of \(t_{1998}\)?
The \(n\)th term of an arithmetic sequence is given by \(t_n=555-7n\). If \(S_n=t_1+t_2+\cdots+t_n\), determine the smallest value of \(n\) for which \(S_n<0\).