Arithmetic Progressions & Sequences

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✖️ 1. Defining arithmetic sequences via recursive definition and finding the common difference

🔁 Recursive Definition & Common Difference

An arithmetic sequence adds the same number to get the next term.
The recursive rule is $a_n = a_{n-1} + d$ where $d$ is the common difference.
To find $d$ , subtract any term from the next term: $d = a_n - a_{n-1}$ .
If $d > 0$ the sequence increases; if $d < 0$ it decreases.
The common difference stays constant throughout the entire sequence.

Example: Sequence 3, 7, 11, 15 has $d = 7 - 3 = 4$ , so $a_n = a_{n-1} + 4$ .

💡 Think: Each step climbs (or drops) by the same fixed amount every time.

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1. Defining arithmetic sequences via recursive definition and finding the common difference

Arithmetic Sequences and the Common Difference

An arithmetic sequence is a sequence where each term after the first is obtained by adding a fixed constant to the previous term. This recursive relationship is expressed as $a_n = a_{n-1} + d$ , where $d$ is the common difference.

Intuition: Each step forward in the sequence involves the same additive jump, creating a uniform pattern of growth or decay.

Core Rules:

The common difference $d = a_n - a_{n-1}$ for any consecutive terms.
If $d > 0$ , the sequence is increasing; if $d < 0$ , it is decreasing; if $d = 0$ , all terms are identical.
The recursive formula requires knowing the previous term to compute the next.
The sequence is fully determined by the first term $a_1$ and the common difference $d$ .

Consequence: The common difference characterizes the entire sequence's behavior and rate of change.

Example: In the sequence $3, 7, 11, 15, \ldots$ , we have $d = 7 - 3 = 4$ , so $a_n = a_{n-1} + 4$ .

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Find the common difference $d$ for the following arithmetic sequence:

$12, 5, -2, -9, \dots$

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✖️ 2. Linking arithmetic sequences to linear models

📈 Connection to Linear Models

An arithmetic sequence is a discrete linear function where $n$ is the input.
The common difference $d$ acts as the slope of the line.
The first term $a_1$ acts as the y-intercept (when $n = 1$ ).
Graphing term number vs. term value produces evenly spaced points on a straight line.
This connects algebra (sequences) to geometry (linear graphs).

Example: Sequence 5, 8, 11, 14 has slope $d = 3$ ; plotting (1,5), (2,8), (3,11), (4,14) forms a line.

💡 Visual: Dots marching up (or down) a perfectly straight staircase.

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2. Linking arithmetic sequences to linear models

Arithmetic Sequences as Linear Functions

An arithmetic sequence corresponds to a discrete linear function where the term number $n$ acts as the independent variable and the term value $a_n$ as the dependent variable. The common difference $d$ is precisely the slope of this linear relationship.

Intuition: Plotting the terms $(n, a_n)$ yields points lying on a straight line, revealing the sequence's linear structure.

Core Rules:

The relationship $a_n = a_1 + (n-1)d$ mirrors the slope-intercept form $y = mx + b$ , where slope $m = d$ .
The slope $d$ measures the rate of change per unit increase in $n$ .
The $y$ -intercept analogue is $a_1 - d$ (the value when $n = 0$ is extrapolated).
This connection allows applying linear modeling techniques to sequence problems.

Consequence: Arithmetic sequences are linear models in discrete settings, enabling graphical and algebraic analysis using familiar tools from coordinate geometry.

Example: For $a_n = 5 + 3(n-1)$ , the slope is $3$ , matching the common difference.

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An arithmetic sequence is defined by the formula $a_n = 7 + 4(n-1)$ .

If this sequence is graphed as a discrete linear function where $n$ is the independent variable, what is the slope of the line?

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✖️ 3. Using the explicit formula for the n-th term

🎯 Explicit Formula for the n-th Term

The explicit formula is $a_n = a_1 + (n - 1)d$ where $a_1$ is the first term.
This lets you jump directly to any term without computing all previous terms.
The factor $(n - 1)$ counts how many steps of size $d$ you take from $a_1$ .
Plug in $n$ , $a_1$ , and $d$ to calculate $a_n$ instantly.
Use this formula when you need a specific term far into the sequence.

Example: For $a_1 = 2$ and $d = 5$ , the 10th term is $a_{10} = 2 + (10 - 1) \cdot 5 = 2 + 45 = 47$ .

💡 Shortcut: Start + (steps × step-size) = destination.

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3. Using the explicit formula for the n-th term

Explicit Formula for the $n$ -th Term

The explicit formula $a_n = a_1 + (n-1)d$ directly computes the $n$ -th term of an arithmetic sequence without requiring prior terms. Here, $a_1$ is the first term, $d$ is the common difference, and $n$ is the term position.

Intuition: To reach the $n$ -th term, we start at $a_1$ and add the common difference $(n-1)$ times, since there are $(n-1)$ steps from term 1 to term $n$ .

Core Rules:

The formula applies for any positive integer $n$ .
Direct computation: No need for recursion; any term is accessible immediately.
Rearranging gives $d = \frac{a_n - a_1}{n - 1}$ when $d$ is unknown.
If $a_1$ and $a_n$ are known along with $n$ , we can solve for $d$ .

Consequence: The explicit formula is computationally efficient and essential for solving problems involving distant terms.

Example: For $a_1 = 2$ , $d = 5$ , the 10th term is $a_{10} = 2 + (10-1) \cdot 5 = 47$ .

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An arithmetic sequence has a first term $a_1 = 5$ and a common difference $d = 3$ .

Calculate the 12th term of this sequence.

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✖️ 4. Defining an arithmetic series and deriving the sum formula

➕ Arithmetic Series & Sum Formula

An arithmetic series is the sum of terms in an arithmetic sequence.
The sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$ .
This formula works by pairing first and last terms, which all sum to the same value.
You can also write it as $S_n = \frac{n}{2}[2a_1 + (n - 1)d]$ using only $a_1$ and $d$ .
Multiply the average of first and last term by the number of terms.

Example: Sum of 2, 5, 8, 11, 14 (5 terms) is $S_5 = \frac{5}{2}(2 + 14) = \frac{5}{2} \cdot 16 = 40$ .

💡 Trick: (First + Last) ÷ 2 × Count = Total.

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4. Defining an arithmetic series and deriving the sum formula

Arithmetic Series and the Sum Formula

An arithmetic series is the sum of the first $n$ terms of an arithmetic sequence. The sum formula is $S_n = \frac{n}{2}(a_1 + a_n)$ , where $S_n$ denotes the sum, $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term.

Intuition: Pairing terms symmetrically from the ends yields $n/2$ pairs, each summing to $a_1 + a_n$ , giving the total sum.

Core Rules:

Alternative form: $S_n = \frac{n}{2}[2a_1 + (n-1)d]$ when $a_n$ is not directly known.
The formula requires knowing $n$ , $a_1$ , and either $a_n$ or $d$ .
Derivation: Write $S_n$ forward and backward, add term-by-term to get $2S_n = n(a_1 + a_n)$ .
The sum grows quadratically in $n$ when $d \neq 0$ .

Consequence: The sum formula enables efficient computation of cumulative totals without adding terms individually.

Example: For $a_1 = 3$ , $a_5 = 15$ , $n = 5$ : $S_5 = \frac{5}{2}(3 + 15) = 45$ .

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Find the sum of the first $10$ terms of an arithmetic sequence where the first term is $5$ and the 10th term is $41$ .

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✖️ 5. Applications in depreciation and kinematics

🛠️ Real-World Applications

Straight-line depreciation: An asset loses the same dollar value each year.
If a machine costs 10000 dollars and depreciates 1000 dollars yearly, value after $n$ years is $10000 - 1000n$ .
Uniform acceleration: Velocity increases by the same amount each second in physics.
If a car starts at 5 m/s and accelerates 2 m/s every second, velocity at second $n$ is $5 + 2(n - 1)$ .
Both scenarios use the explicit formula $a_n = a_1 + (n - 1)d$ with real units.

Example: A laptop worth 1200 dollars depreciates 150 dollars/year; after 4 years: $1200 - 150 \cdot 4 = 600$ dollars.

💡 Remember: Constant change over time = arithmetic progression in action.

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5. Applications in depreciation and kinematics

Applications: Depreciation and Uniform Acceleration

Arithmetic progressions model real-world scenarios involving constant rate changes. In accounting, straight-line depreciation reduces an asset's value by a fixed amount $d$ each period. In kinematics, uniform acceleration changes velocity by a constant increment per time unit.

Intuition: When a quantity changes by the same amount repeatedly, an arithmetic sequence naturally describes its evolution over time.

Core Rules:

Depreciation: If an asset starts at value $V_0$ and depreciates by $d$ dollars annually, its value after $n$ years is $V_n = V_0 - nd$ .
Kinematics: With initial velocity $v_0$ and constant acceleration $a$ , velocity at time $t_n$ is $v_n = v_0 + na$ (discrete time steps).
Both applications use the explicit formula structure $a_n = a_1 + (n-1)d$ .
The common difference represents the rate of change (depreciation rate or acceleration).

Consequence: Recognizing arithmetic patterns simplifies financial and physical predictions.

Example: An asset worth 10000 dollars depreciating 500 dollars yearly has value $10000 - 5 \cdot 500 = 7500$ dollars after 5 years.

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A delivery van is purchased for 35000 dollars. It undergoes straight-line depreciation, losing 2500 dollars in value each year. What is the value of the van, in dollars, after 4 years?

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✖️ 1. Defining arithmetic sequences via recursive definition and finding the common difference

🔁 Recursive Definition & Common Difference

1. Defining arithmetic sequences via recursive definition and finding the common difference

Arithmetic Sequences and the Common Difference

✖️ 2. Linking arithmetic sequences to linear models

📈 Connection to Linear Models

2. Linking arithmetic sequences to linear models

Arithmetic Sequences as Linear Functions

✖️ 3. Using the explicit formula for the n-th term

🎯 Explicit Formula for the n-th Term

3. Using the explicit formula for the n-th term

Explicit Formula for the nnn-th Term

✖️ 4. Defining an arithmetic series and deriving the sum formula

➕ Arithmetic Series & Sum Formula

4. Defining an arithmetic series and deriving the sum formula

Arithmetic Series and the Sum Formula

✖️ 5. Applications in depreciation and kinematics

🛠️ Real-World Applications

5. Applications in depreciation and kinematics

Applications: Depreciation and Uniform Acceleration

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Explicit Formula for the $n$ -th Term