An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant value to the preceding term. This constant difference between consecutive terms is known as the common difference, denoted by \(d\). Arithmetic progressions are widely applied in modeling uniform additions or deductions, such as layered building steps, seating capacities across auditorium rows, uniform salary increments, or daily work production.
Illustrate why \(S_n = \dfrac{n}{2}[2a + (n - 1)d]\) by combining two staircase-shaped stair-step bar diagrams to form a single solid rectangle.
An arithmetic progression represents steady, linear change. The key characteristics are the first term \(a\) and the uniform common difference \(d\).
A sequence is a set of numbers arranged according to a specific pattern. Each number in a sequence is called a term, written as \(T_1, T_2, T_3, \ldots, T_n\). A finite sequence has a limited number of terms, while an infinite sequence continues indefinitely. An arithmetic progression is a sequence where each subsequent term is obtained by adding a fixed constant to the preceding term. The common difference is \(d = T_2 - T_1 = T_3 - T_2 = \ldots = T_n - T_{n-1}\). Note: \(d \neq T_1 - T_2 \neq T_2 - T_3 \neq \ldots \neq T_{n-1} - T_n\). The subtraction must always be a succeeding term minus the preceding term.
Let the first term be \(a\) (\(T_1 = a\)) and the common difference be \(d\). Then \(T_1 = a = a + 0d\), \(T_2 = a + d = a + 1d\), \(T_3 = (a + d) + d = a + 2d\). By deduction, the formula for the \(n^{\text{th}}\) term is \(T_n = a + (n - 1)d\), where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms (\(n \in \mathbb{Z}^+\)).
An arithmetic series is \(T_1 + T_2 + T_3 + \ldots + T_n\). By representing terms as grid heights and combining two identical sets of grids, a uniform rectangle of width \(n\) and height \([2a + (n - 1)d]\) is formed. Since \(S_n\) represents half of this combined area: \(S_n = \dfrac{n}{2}[2a + (n - 1)d]\). Using the last term \(l = T_n = a + (n - 1)d\), the sum can also be written as \(S_n = \dfrac{n}{2}[a + T_n]\) or \(S_n = \dfrac{n}{2}[a + l]\).
The \(n^{\text{th}}\) term can be found by subtracting the sum of the first \((n - 1)\) terms from the sum of the first \(n\) terms: \(T_n = S_n - S_{n-1}\). Similarly, the sum of a specific block of consecutive terms can be evaluated using sums from the start. For example, \(S_8 - S_5 = T_6 + T_7 + T_8\), and the sum of 200 terms after the first 50 terms is given by \(S_{250} - S_{50}\).
Understanding the problem: Identify the given values (\(a\), \(d\), \(T_n\), or \(S_n\)) and what needs to be determined.
Planning a strategy: Formulate the arithmetic progression and choose the appropriate formula (\(T_n\) or \(S_n\)).
Implementing the strategy: Solve algebraically or solve quadratic inequalities where \(n\) must be a positive integer (\(n \in \mathbb{Z}^+\)).
Making a conclusion: State the final result in the context of the question.
\(n^{\text{th}}\) term of an arithmetic progression
Use this to find any term directly without listing every preceding number.
Sum of the first \(n\) terms
Pairing the first and last terms yields identical sums.
Answer in your mind, then press "Check Answer".
Determine whether the sequence \(5 - x, 5, 5 + x, \ldots\) is an arithmetic progression and state its common difference.
Find the \(15^{\text{th}}\) term of the arithmetic progression \(-4, 2, 8, \ldots\).
The sum of the first \(n\) terms of an arithmetic progression is given by \(S_n = 2n^2 - 5n\). Find the first term, \(T_1\), and the \(9^{\text{th}}\) term, \(T_9\).
Assuming that a sequence is an arithmetic progression just because the first two terms have a difference, without checking subsequent pairs.
You must verify that the difference between consecutive pairs is equal, for example checking \(T_2 - T_1\) and \(T_3 - T_2\).
Believing that the common difference \(d\) must always be positive.
\(d\) can be negative for decreasing sequences, zero for constant sequences, or a fraction.
Thinking that \(T_n\) equals \(\dfrac{S_n}{n}\).
The correct relationship between terms and sums is \(T_n = S_n - S_{n-1}\).
Mistakenly assuming that finding the sum from term \(p\) to term \(q\) is \(S_q - S_p\).
The correct expression is \(S_q - S_{p-1}\).
Objective: Identify arithmetic progressions, use the \(n^{\text{th}}\) term and sum formulas, and match key concepts.
1 What is the common difference, \(d\), of the arithmetic progression \(-35, -21, -7, \ldots\)?
2 Given an arithmetic progression with first term \(a = 10\) and common difference \(d = -4\), what is the value of the \(7^{\text{th}}\) term, \(T_7\)?
3 In general, the \(n^{\text{th}}\) term of an arithmetic progression can be derived from the sum of its terms using the formula \(T_n = S_n - \) ________.
4 When the first term is \(a\) and the last term is \(l\), the sum of the first \(n\) terms can be written as \(S_n = \dfrac{n}{2}[\) ________ \(]\).
Match each arithmetic progression concept or problem statement with its correct mathematical expression or value. On a phone, tap a term first, then tap the matching box.
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