General term of a sequence
Ximena pays for a music app and the fee goes up every month; Diego drops a ball and sees that each bounce is lower. Can all of that be described with a single formula that gives any term without calculating the previous ones? That is the general term, and you are going to build it on this page.
The general term
$$a_n = \frac{2n-1}{n+3} \quad\Rightarrow\quad a_1 = \frac{1}{4},\ a_2 = \frac{3}{5},\ a_3 = \frac{5}{6},\ a_4 = \frac{7}{7} = 1,\ldots$$Type an expression and watch how the bars change as $n$ increases:
If the sequence is arithmetic or geometric, you already know the formulas: $a_n = a_1 + (n-1)d$ and $a_n = a_1 \cdot r^{\,n-1}$. For example, $5, 9, 13, \ldots$ has $d=4$, so $a_n = 5 + 4(n-1) = 4n+1$. And $3, -6, 12, \ldots$ has $r=-2$, so $a_n = 3(-2)^{n-1}$.
When it is neither of the two, look for a relationship with $n$. For example, in $0, 3, 8, 15, 24, \ldots$ each term is a perfect square minus 1: $a_n = n^2 - 1$. Check it: $n=1$ gives $0$, $n=2$ gives $3$, $n=3$ gives $8$.
Given the sequence whose nth term is $a_n = \frac{2n-1}{n+3}$, calculate its first five terms.
Given the sequence whose nth term is $a_n = \frac{2n-1}{n+3}$, calculate its first five terms.
Find the expression for the nth term of the following arithmetic sequence: $5, 9, 13, 17, 21, \dots$
Find the expression for the nth term of the following arithmetic sequence:
$5, 9, 13, 17, 21, \dots$
Find the expression for the nth term of the following geometric sequence: $3, -6, 12, -24, 48, \dots$
Find the expression for the nth term of the following geometric sequence:
$3, -6, 12, -24, 48, \dots$
If the nth term of a sequence is $b_n = n^2 - 5n + 6$, find the exact value of the twentieth term ($b_{20}$).
If the nth term of a sequence is $b_n = n^2 - 5n + 6$, find the exact value of the twentieth term ($b_{20}$).
Check mathematically whether the number $47$ belongs to the sequence with nth term $c_n = 3n + 2$. If so, state its position in the sequence.
Check mathematically whether the number $47$ belongs to the sequence with nth term $c_n = 3n + 2$. If so, state its position in the sequence.
Find the general term of the sequence $0, 3, 8, 15, 24, \dots$ and use it to calculate the exact value of the fiftieth term ($a_{50}$).
Find the general term of the sequence $0, 3, 8, 15, 24, \dots$ and use it to calculate the exact value of the fiftieth term ($a_{50}$).
Given the sequence whose general term is $a_n = \frac{n^2+1}{n}$, find the expression for the general term of the difference sequence defined as $d_n = a_{n+1} - a_n$…
Given the sequence whose general term is $a_n = \frac{n^2+1}{n}$, find the expression for the general term of the difference sequence defined as $d_n = a_{n+1} - a_n$ and simplify the result as much as possible.
Numerator and denominator separately
When the general term is a fraction, study numerators and denominators as two independent sequences and combine them at the end.
Write down the nth term $f_n$ of the following sequence of fractions: $\frac{1}{2}, \frac{4}{5}, \frac{9}{10}, \frac{16}{17}, \dots$
Write down the nth term $f_n$ of the following sequence of fractions:
$\frac{1}{2}, \frac{4}{5}, \frac{9}{10}, \frac{16}{17}, \dots$
Find the nth term $a_n$ of the following sequence by combining the patterns governing its numerator and denominator independently:…
Find the nth term $a_n$ of the following sequence by combining the patterns governing its numerator and denominator independently:
$\frac{2}{3}, \frac{5}{9}, \frac{8}{27}, \frac{11}{81}, \dots$
Alternating signs
The factor $(-1)^n$ changes the sign at every term: negative if $n$ is odd and positive if $n$ is even (with $(-1)^{n+1}$ it is the other way around).
Even and odd positions with different rules
In $1, 4, 3, 8, 5, 12, 7, 16, \ldots$ the odd positions give $1, 3, 5, 7$ and the even positions give $4, 8, 12, 16$. Separate the two sequences and look for the pattern in each:
-
Position $n$ odd: the value is $n$.
-
Position $n$ even: the value is $2n$.
To unify them you need something that equals $0$ in one case and $1$ in the other. The expression $\dfrac{1+(-1)^n}{2}$ equals $1$ if $n$ is even and $0$ if it is odd. Then:
$$a_n = n + \frac{1+(-1)^n}{2}\, n = n\cdot\frac{3+(-1)^n}{2}$$Check: $n=1$ gives $1\cdot\frac{2}{2}=1$; $n=2$ gives $2\cdot\frac{4}{2}=4$; $n=3$ gives $3$; $n=4$ gives $8$ ✓.
Write down the nth term of a sequence whose terms are consecutive integers with alternating signs: $-1, 2, -3, 4, -5, \dots$
Write down the nth term of a sequence whose terms are consecutive integers with alternating signs:
$-1, 2, -3, 4, -5, \dots$
Find a single explicit general term $a_n$ for the sequence $1, 4, 3, 8, 5, 12, 7, 16, \dots$ using the expression $(-1)^n$ to unify the differing behavior shown by the…
Find a single explicit general term $a_n$ for the sequence $1, 4, 3, 8, 5, 12, 7, 16, \dots$ using the expression $(-1)^n$ to unify the differing behavior shown by the even and odd terms.
Recurrence relations
A sequence can also be given by recurrence: each term is calculated from the previous ones.
$$a_1 = 2, \qquad a_n = 3a_{n-1} - 1 \ (n \ge 2) \quad\Rightarrow\quad 2,\ 5,\ 14,\ 41,\ \ldots$$Many situations are naturally written this way: an amount of money that is multiplied by a factor each year and then has a fixed amount subtracted satisfies $C_n = k\,C_{n-1} - c$, with $C_0$ given.
Another case: $x_1=1$, $x_2=2$, $x_n = x_{n-1}+2x_{n-2}$ gives $1, 2, 4, 8, 16, 32$. The terms are powers of $2$, so $x_n = 2^{n-1}$.
Given the recurrence relation $a_1 = 2$ and $a_n = 3a_{n-1} - 1$ for $n \ge 2$, calculate the first four terms of the sequence.
Given the recurrence relation $a_1 = 2$ and $a_n = 3a_{n-1} - 1$ for $n \ge 2$, calculate the first four terms of the sequence.
A recurrence sequence is defined by the initial conditions $x_1 = 1$, $x_2 = 2$ and the recurrence relation $x_n = x_{n-1} + 2x_{n-2}$ for all $n \ge 3$. Calculate the…
A recurrence sequence is defined by the initial conditions $x_1 = 1$, $x_2 = 2$ and the recurrence relation $x_n = x_{n-1} + 2x_{n-2}$ for all $n \ge 3$. Calculate the terms of the sequence up to $x_6$ and deduce an explicit formula for the general term $x_n$ in terms of powers of $2$.
The sequence $a_n$ is defined by the recurrence relation $a_1 = 3$ and $a_n = 2a_{n-1} + 1$ for $n \ge 2$. To find its explicit general term, define a new auxiliary…
The sequence $a_n$ is defined by the recurrence relation $a_1 = 3$ and $a_n = 2a_{n-1} + 1$ for $n \ge 2$. To find its explicit general term, define a new auxiliary sequence $b_n = a_n + 1$. Prove that $b_n$ is a geometric progression, find its general term $b_n$ and, from it, deduce the general term of $a_n$.
Sequences that model real situations
Populations, depreciation, training, or savings are described with sequences. First identify whether at each step a fixed amount is added or multiplied, and whether $n$ starts at $0$ or at $1$.
If the pattern is "a fixed amount is added," the sequence is arithmetic. For example, with toothpick figures where each new figure adds 5 toothpicks and the first uses 6 (6, 11, 16, …): $P_n = 6 + 5(n-1) = 5n+1$.
When the question asks "when does it exceed or drop below a certain value?", set up an inequality with the general term. If $n$ is in an exponent, use logarithms: for example, $10\,(0.75)^n < 1$ gives $n > \dfrac{\ln 0.1}{\ln 0.75} \approx 8.0$, so 9 bounces are needed. Careful: when dividing by $\ln 0.75$, which is negative, the inequality flips.
A population of microscopic algae in a controlled lake triples its covered surface area each week. If in the first week (week $n=1$) they cover a surface area of…
A population of microscopic algae in a controlled lake triples its covered surface area each week. If in the first week (week $n=1$) they cover a surface area of $2 \text{ m}^2$:
- a) Write down the nth term $S_n$ representing the covered surface area after $n$ weeks.
- b) Calculate how many full weeks must pass for the covered surface area to exceed $1000 \text{ m}^2$ for the first time.
Observe the following sequence of figures made up of adjacent hexagons sharing one or more sides. Figure 1 is made of 6 matchsticks, Figure 2 of 11 matchsticks and…
Observe the following sequence of figures made up of adjacent hexagons sharing one or more sides. Figure 1 is made of 6 matchsticks, Figure 2 of 11 matchsticks and Figure 3 of 16 matchsticks. Write down the nth term for the number of matchsticks $P_n$ needed to build Figure $n$.
The market value of a new car is $480\,000$ pesos. It is estimated that each year it loses $15\,\%$ of its value compared to the immediately preceding year. Find the nth…
The market value of a new car is $480\,000$ pesos. It is estimated that each year it loses $15\,\%$ of its value compared to the immediately preceding year. Find the nth term $V_n$ expressing the value of the car after $n$ years (considering $n=0$ to be the time of purchase), and determine its estimated value after $5$ years.
An amateur athlete is preparing for a half marathon. On the first day of training, he runs $5$ km, and decides to increase the daily distance by exactly $500$ meters…
An amateur athlete is preparing for a half marathon. On the first day of training, he runs $5$ km, and decides to increase the daily distance by exactly $500$ meters compared to the previous day. Express the general term $d_n$ for the distance in kilometers run on day $n$. On which day of his training will he run exactly $15$ km?
A rubber ball is dropped from an initial height of $10$ meters. On each bounce against the ground, the ball reaches a height equal to $\frac{3}{4}$ of the maximum height…
A rubber ball is dropped from an initial height of $10$ meters. On each bounce against the ground, the ball reaches a height equal to $\frac{3}{4}$ of the maximum height reached on the immediately preceding bounce. Write down the general term $h_n$ for the height reached after the $n$th bounce. Using logarithms, determine after how many bounces the maximum height will be less than $1$ meter for the first time.
Extension: geometric patterns, monotonicity, and boundedness
Two common patterns. In a network of $n$ computers where every computer is connected to every other, the number of connections is $C_n=\dfrac{n(n-1)}{2}$ (terms $0, 1, 3, 6, \ldots$). In a triangle fractal, if each step multiplies the number of new triangles by 3 and divides the area of each one by 4, you have $T_n = 3^{n-1}$ and $A_n = \dfrac{A}{4^n}$.
Increasing or decreasing
To check whether a sequence increases or decreases, study the sign of $a_{n+1}-a_n$. For example, in $a_n=\dfrac{3n-1}{n+2}$:
$$a_{n+1}-a_n=\frac{3n+2}{n+3}-\frac{3n-1}{n+2}=\frac{(3n+2)(n+2)-(3n-1)(n+3)}{(n+3)(n+2)}=\frac{7}{(n+3)(n+2)}>0$$so it is strictly increasing.
Boundedness
A sequence is bounded below by $m$ if $a_n \ge m$ for all $n$, and bounded above by $M$ if $a_n \le M$ for all $n$. To prove it, set up the inequality with $a_n$ and work it out until you reach something obviously true. Since the denominators are positive, you can cross-multiply without flipping the inequality.
For $a_n=\dfrac{5n+2}{2n+1}$ the procedure is the same: $\dfrac{5n+2}{2n+1}\ge\dfrac{7}{3}$ is equivalent to $15n+6\ge 14n+7$, that is $n\ge 1$ (true); and $\dfrac{5n+2}{2n+1}\le\dfrac{5}{2}$ is equivalent to $10n+4\le 10n+5$, that is $4\le 5$ (true).
A geometric fractal is constructed starting from an equilateral triangle of area $A$. At each step, the midpoints of the sides of the unshaded triangles are joined to…
A geometric fractal is constructed starting from an equilateral triangle of area $A$. At each step, the midpoints of the sides of the unshaded triangles are joined to form 4 smaller triangles, and only the central one is shaded. Find the general term $T_n$ expressing the number of new shaded triangles added at step $n$, and the general term $A_n$ for the area of a single one of those new triangles added at step $n$.
Prove mathematically, by studying the algebraic sign of the difference $a_{n+1} - a_n$, whether the sequence $a_n = \frac{3n-1}{n+2}$ is strictly increasing or strictly…
Prove mathematically, by studying the algebraic sign of the difference $a_{n+1} - a_n$, whether the sequence $a_n = \frac{3n-1}{n+2}$ is strictly increasing or strictly decreasing.
In a computer network, the total number of possible direct connections between $n$ computers (if all are physically connected to each other) forms a sequence $C_n$.…
In a computer network, the total number of possible direct connections between $n$ computers (if all are physically connected to each other) forms a sequence $C_n$. Knowing that $C_1 = 0$, $C_2 = 1$, $C_3 = 3$, and $C_4 = 6$, determine the general term $C_n$. From it, deduce how many computers make up a network that has exactly $1225$ direct connections.
Study the boundedness of the sequence $a_n = \frac{5n+2}{2n+1}$. To do so, prove algebraically that the sequence is bounded below by $\frac{7}{3}$ and bounded above by…
Study the boundedness of the sequence $a_n = \frac{5n+2}{2n+1}$. To do so, prove algebraically that the sequence is bounded below by $\frac{7}{3}$ and bounded above by $\frac{5}{2}$.
Go back to the beginning: the app's increasing fee is an arithmetic sequence and the ball's bounces are a geometric one; in both cases a formula in $n$ gives any term without calculating the previous ones.