Observe the waxing and waning of the moon from new moon to full moon, or Wang Fang’s annual height records from age 1 to 17. These data are not random; they are arranged in chronological order. In mathematics, sucha sequence of numbers arranged in a definite order, helps us capture the evolution patterns in discrete systems. This is a sequence—a crucial mathematical model for describing dynamic patterns.
Definition and Core Characteristics of Sequences
At its core, a sequence is a special type of function where the independent variable is the term’s "position" or "index" $n$, and the dependent variable is the value $a_n$ at that position. Throughgeneral term formula, we can predict any term in the sequence just as we would use a function expression.
Key Elements:
- Order: Terms in a sequence must be arranged in a definite order; changing the order results in a different sequence.
- Discreteness: The domain is the set of positive integers $\mathbb{N}^*$ or a finite subset thereof, so its graph consists of isolated points on the coordinate plane.
- Correspondence: There exists a definite functional relationship between the $n$th term $a_n$ and its index $n$, expressed as $a_n = f(n)$.
A sequence is a special kind of function. If the relationship between the $n$th term $a_n$ and its index $n$ in sequence $\{a_n\}$ can be expressed by a single formula, this formula is called thegeneral term formula.
$$a_1, a_2, a_3, \dots, a_n, \dots \quad \text{abbreviated as} \ \{a_n\}$$
1. Gather polynomial terms: one $x^2$ square, three $x$ rectangular strips, and two $1 \times 1$ unit squares.
2. Begin assembling them geometrically.
3. They perfectly form a larger continuous rectangle! Width is $(x+2)$, height is $(x+1)$.
QUESTION 1
Which of the following statements about sequences is correct?
The sequences $1, 2, 3, 4$ and $4, 3, 2, 1$ are the same sequence
Terms in a sequence cannot repeat
A sequence can be viewed as a function with domain being the set of positive integers (or a subset thereof)
The graph of a sequence is a continuous line or curve
Correct!
The essence of a sequence lies in its "definite order," and since its domain is discrete positive integers, its graph consists of isolated points.
Incorrect
Please note the definition of a sequence: a list of numbers arranged in a "definite order." Changing the order changes the sequence.
QUESTION 2
Given the first four terms of a sequence: $1, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{4}, \dots$, which of the following could be its general term formula?
$a_n = \frac{(-1)^n}{n}$
$a_n = \frac{(-1)^{n+1}}{n}$
$a_n = \frac{1}{n}$
$a_n = (-1)^n \cdot n$
Perfect!
The first term $a_1=1$ is positive, so the sign factor should be $(-1)^{1+1}$, and the denominator increases with $n$. The general term is $a_n = \frac{(-1)^{n+1}}{n}$.
Hint
Pay attention to whether the first term is positive or negative. When $n=1$, $(-1)^n$ yields $-1$, while $(-1)^{n+1}$ yields $1$.
QUESTION 3
If the general term formula of sequence $\{a_n\}$ is $a_n = n^2 + 2n$, then which term is $120$?
The 12th term
The 10th term
The 8th term
Not a term of this sequence
Correct calculation!
Set $n^2 + 2n = 120$, i.e., $n^2 + 2n - 120 = 0$. Solving gives $n=10$ or $n=-12$ (discarded). Thus, it is the 10th term.
Hint
Solve the equation $n^2 + 2n = 120$. Remember that $n$ must be a positive integer!
QUESTION 4
In the Sierpinski triangle, as the iteration count $n$ increases, the number of colored triangles follows $1, 3, 9, 27 \dots$. What is the number of colored triangles in the $n$th figure?
$3n$
$3^n$
$3^{n-1}$
$n^3$
Sharp observation!
This is a geometric growth pattern: $3^0, 3^1, 3^2, 3^3 \dots$, corresponding to indices $n=1, 2, 3, 4 \dots$, so the general term is $3^{n-1}$.
Incorrect
Check if the formula equals $1$ when $n=1$. $3^1=3$, but $3^{1-1}=1$.
QUESTION 5
One possible general term formula for the sequence $2, 0, 2, 0, \dots$ is:
$a_n = (-1)^{n+1} + 1$
$a_n = (-1)^n + 1$
$a_n = \cos(n\pi)$
$a_n = 2n - 2$
Correct!
When $n$ is odd, $a_n=1+1=2$; when $n$ is even, $a_n=-1+1=0$.
Hint
This is an oscillating sequence. Use the parity property of $(-1)^n$ to construct cancellation or summation of constant terms.
QUESTION 6
If each term starting from the second is greater than its preceding term, the sequence is called:
a finite sequence
an increasing sequence
a decreasing sequence
a constant sequence
Correct!
This is the strict definition of an increasing sequence: $a_n > a_{n-1}$.
Incorrect
"Greater than" corresponds to "increasing", "less than" to "decreasing", and "equal" to "constant".
QUESTION 7
Given the general term formula of sequence $\{a_n\}$ is $a_n = \frac{n^2+n}{2}$, what is $a_5$?
10
15
20
25
Correct!
$a_5 = \frac{5^2 + 5}{2} = \frac{30}{2} = 15$.
Hint
Simply substitute $n=5$ into the formula to calculate.
QUESTION 8
What characteristic does the general term formula $a_n = (-1)^n$ of the sequence $-1, 1, -1, 1, \dots$ represent?
It is an increasing sequence
It is a decreasing sequence
It is an oscillating sequence
It is a finite sequence
Exactly!
The values alternate between positive and negative.
Incorrect
Observing the values: $-1, 1, -1, 1$, it neither consistently increases nor decreases.
QUESTION 9
Can a sequence have an infinite number of terms?
Yes, it is called an infinite sequence
No, sequences must have an endpoint
Only constant sequences can be infinite
Only arithmetic sequences can be infinite
Correct!
Sequences with infinitely many terms are called infinite sequences, such as the natural number sequence.
Incorrect
By definition, sequences with a finite number of terms are finite sequences, and those with an infinite number are infinite sequences.
Challenge: Logic and Modeling of Sequences
From Discrete Patterns to Rigorous Proof
Task 1
Write the first 10 terms of the following sequences and sketch their graphs: (1) The sequence formed by arranging reciprocals of all positive integers in ascending order; (2) The sequence formed by the values of function $f(x) = 2x + 1$ as the independent variable $x$ takes values $1, 2, 3, \dots$; (3) $a_n = \begin{cases} 2, & n \text{ is odd} \\ n+1, & n \text{ is even} \end{cases}$
Reference Answer:
(1) $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}$. The graph consists of isolated points on the curve of the inverse proportion function in the first quadrant.
(2) $3, 5, 7, 9, 11, 13, 15, 17, 19, 21$. The graph consists of a series of points on a straight line with slope 2.
(3) $2, 3, 2, 5, 2, 7, 2, 9, 2, 11$. The graph shows odd-numbered terms on the line $y=2$, and even-numbered terms on the line $y=x+1$.
(1) $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{10}$. The graph consists of isolated points on the curve of the inverse proportion function in the first quadrant.
(2) $3, 5, 7, 9, 11, 13, 15, 17, 19, 21$. The graph consists of a series of points on a straight line with slope 2.
(3) $2, 3, 2, 5, 2, 7, 2, 9, 2, 11$. The graph shows odd-numbered terms on the line $y=2$, and even-numbered terms on the line $y=x+1$.
Task 2
Given sequence $\{a_n\}$ with initial term $a_1=1$ and recursive formula $a_n = 1 + \frac{1}{a_{n-1}}$ for $n \ge 2$, write down the first 5 terms.
Reference Answer:
$a_1 = 1$
$a_2 = 1 + \frac{1}{1} = 2$
$a_3 = 1 + \frac{1}{2} = \frac{3}{2}$
$a_4 = 1 + \frac{1}{3/2} = 1 + \frac{2}{3} = \frac{5}{3}$
$a_5 = 1 + \frac{1}{5/3} = 1 + \frac{3}{5} = \frac{8}{5}$
The first 5 terms are: $1, 2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}$.
$a_1 = 1$
$a_2 = 1 + \frac{1}{1} = 2$
$a_3 = 1 + \frac{1}{2} = \frac{3}{2}$
$a_4 = 1 + \frac{1}{3/2} = 1 + \frac{2}{3} = \frac{5}{3}$
$a_5 = 1 + \frac{1}{5/3} = 1 + \frac{3}{5} = \frac{8}{5}$
The first 5 terms are: $1, 2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}$.
Task 3
Observe the characteristics of the following sequence and fill in the blanks with appropriate numbers: $(\quad), -4, 9, (\quad), 25, (\quad), 49$, and write a general term formula.
Reference Answer:
Observing closely, the absolute values are $n^2$, and signs alternate. Terms 2, 4, and 6 are negative.
Fill in the blanks:$1$, -4, 9, $-16$, 25, $-36$, 49.
General term formula: $a_n = (-1)^{n+1} \cdot n^2$.
Observing closely, the absolute values are $n^2$, and signs alternate. Terms 2, 4, and 6 are negative.
Fill in the blanks:$1$, -4, 9, $-16$, 25, $-36$, 49.
General term formula: $a_n = (-1)^{n+1} \cdot n^2$.
Task 4
Given sequences $\{a_n\}$ and $\{b_n\}$ are both arithmetic sequences with common differences $d_1$ and $d_2$. If $c_n = a_n + 2b_n$, (1) Is $\{c_n\}$ an arithmetic sequence? (2) If $d_1=d_2=2$ and $a_1=b_1=1$, find the general term of $\{c_n\}$.
Reference Answer:
(1) Yes. $c_{n+1}-c_n = (a_{n+1}-a_n) + 2(b_{n+1}-b_n) = d_1 + 2d_2$, a constant. Hence, $\{c_n\}$ is an arithmetic sequence.
(2) $c_1 = a_1 + 2b_1 = 3$. New common difference $d = d_1 + 2d_2 = 2 + 2(2) = 6$. General term formula: $c_n = 3 + (n-1)6 = 6n - 3$.
(1) Yes. $c_{n+1}-c_n = (a_{n+1}-a_n) + 2(b_{n+1}-b_n) = d_1 + 2d_2$, a constant. Hence, $\{c_n\}$ is an arithmetic sequence.
(2) $c_1 = a_1 + 2b_1 = 3$. New common difference $d = d_1 + 2d_2 = 2 + 2(2) = 6$. General term formula: $c_n = 3 + (n-1)6 = 6n - 3$.
Task 5
Given an arithmetic sequence $\{a_n\}$ with common difference $d$, prove that $\frac{a_m - a_n}{m-n} = d$. Can you explain this result from the perspective of the slope of a line?
Reference Answer:
Proof: $a_m = a_1 + (m-1)d$, $a_n = a_1 + (n-1)d$. Then $a_m - a_n = (m-n)d$. Since $m \neq n$, dividing both sides by $m-n$ gives $\frac{a_m-a_n}{m-n} = d$.
Geometric interpretation:The terms of the sequence lie on the line $y = dx + (a_1-d)$. $\frac{a_m-a_n}{m-n}$ exactly represents the slope formula of the line passing through points $(m, a_m)$ and $(n, a_n)$, whose slope is always equal to the common difference $d$.
Proof: $a_m = a_1 + (m-1)d$, $a_n = a_1 + (n-1)d$. Then $a_m - a_n = (m-n)d$. Since $m \neq n$, dividing both sides by $m-n$ gives $\frac{a_m-a_n}{m-n} = d$.
Geometric interpretation:The terms of the sequence lie on the line $y = dx + (a_1-d)$. $\frac{a_m-a_n}{m-n}$ exactly represents the slope formula of the line passing through points $(m, a_m)$ and $(n, a_n)$, whose slope is always equal to the common difference $d$.
Task 6
When using mathematical induction to prove the formula for the sum of the first $n$ terms of an arithmetic sequence, $S_n = \frac{n(a_1+a_n)}{2}$, if an error occurs when proving from $n=k$ to $n=k+1$, where is the mistake usually made?
Reference Answer:
Common errors include: (1) failing to use the assumption for $n=k$ and instead directly using the conclusion; (2) incorrectly substituting the properties of the general term of an arithmetic sequence in the transition $S_{k+1} = S_k + a_{k+1}$; (3) overlooking the base verification step for $n=1$.
Common errors include: (1) failing to use the assumption for $n=k$ and instead directly using the conclusion; (2) incorrectly substituting the properties of the general term of an arithmetic sequence in the transition $S_{k+1} = S_k + a_{k+1}$; (3) overlooking the base verification step for $n=1$.
Task 7
In the Koch snowflake constructed by Swedish mathematician Koch, if the original equilateral triangle (Figure ①) has side length 1, denoted as perimeter $C_1$. Each step divides each edge into thirds and constructs small equilateral triangles outward. Find $C_4$.
Reference Answer:
$C_1 = 3$. With each iteration, the number of edges becomes 4 times the previous amount, and the length of each edge becomes $1/3$ of the previous length. Therefore, the perimeter becomes $4/3$ times the previous value.
$C_n = 3 \cdot (\frac{4}{3})^{n-1}$.
$C_4 = 3 \cdot (\frac{4}{3})^3 = 3 \cdot \frac{64}{27} = \frac{64}{9}$.
$C_1 = 3$. With each iteration, the number of edges becomes 4 times the previous amount, and the length of each edge becomes $1/3$ of the previous length. Therefore, the perimeter becomes $4/3$ times the previous value.
$C_n = 3 \cdot (\frac{4}{3})^{n-1}$.
$C_4 = 3 \cdot (\frac{4}{3})^3 = 3 \cdot \frac{64}{27} = \frac{64}{9}$.
Task 8
After $t\,s$ since launch, the rocket's height is $h(t)=0.9t^2$. Find: (1) average velocity within $1 \le t \le 2$; (2) instantaneous velocity at $10\,s$. Consider how heights at discrete time points form a sequence.
Reference Answer:
(1) Average velocity $v = \frac{h(2)-h(1)}{2-1} = 0.9(4-1) = 2.7$ m/s.
(2) Instantaneous velocity is the derivative $h'(t) = 1.8t$. At $t=10$, $v = 18$ m/s.
Sequence connection:If we only observe heights at integer seconds $h(1), h(2), \dots, h(n)$, they form a sequence with general term $a_n = 0.9n^2$.
(1) Average velocity $v = \frac{h(2)-h(1)}{2-1} = 0.9(4-1) = 2.7$ m/s.
(2) Instantaneous velocity is the derivative $h'(t) = 1.8t$. At $t=10$, $v = 18$ m/s.
Sequence connection:If we only observe heights at integer seconds $h(1), h(2), \dots, h(n)$, they form a sequence with general term $a_n = 0.9n^2$.
✨ Key Points
Numbers in order,Order comes first.Discrete functions,Points connected in heart.General term formula,Find the right $n$ value.Growth and decrease,Seeking patterns!
💡 Difference Between Sequences and Functions
Although a sequence is a special type of function, its graph consists of discrete points and cannot be connected by a continuous line. Terms are only defined when $n$ is a positive integer.
💡 Make Good Use of Index $n$
The index $n$ starts from $1$. When writing a general term formula, always substitute $n=1$ to verify the first term is correct.
💡 Observe Sign Changes
$(-1)^n$ or $(-1)^{n+1}$ often represent alternating positive and negative patterns. Choose the former if the first term is negative; choose the latter if it is positive.
💡 General Term Formula Is Not Unique
Multiple general term formulas may correspond to the first few terms of the same sequence unless specified otherwise. For example, $1, 2, 4 \dots$ could be $2^{n-1}$, or a complex quadratic polynomial.
💡 Recurrence vs. General Term
The general term formula directly gives the relationship between $n$ and $a_n$, while the recurrence formula gives the relationship between $a_n$ and $a_{n-1}$. When computing values, the general term formula is usually more direct.