Arithmetic Sequence Made Easy

 Arithmetic Sequence

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An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the "common difference." In simpler terms, each term in an arithmetic sequence is obtained by adding the same fixed value to the previous term.

Arithmetic sequences are used in various mathematical and real-world contexts, such as in finance, physics, and computer science. They have a straightforward pattern that makes them easy to understand and work with, and they're often used to model situations where the terms increase or decrease by a consistent amount over time or iterations.

Arithmetic sequences have various practical applications across different fields due to their simple and predictable nature. Here are some common uses of arithmetic sequences:

Several uses of Arithmetic Sequence in:

1. Finance and Economics: Arithmetic sequences play a role in financial planning, helping us figure out things like how our savings grow over time or how we'll be paying off loans.

2. Mathematics Education: In math class, arithmetic sequences are introduced to help us see patterns in numbers. They're like building blocks for understanding more complicated math ideas.

3. Physics and Engineering: Think of arithmetic sequences as handy tools in physics for describing stuff that changes at a constant rate, like how something moves when it's getting faster and faster.

4. Computer Science: When computers follow a specific pattern or rhythm, like counting by the same steps, that's like an arithmetic sequence. It's used in programming and making software work smoothly.

5. Real Estate: If house prices are rising or falling by a fixed amount each year, you can use arithmetic sequences to predict what prices might be in the future.

6. Time and Scheduling: Whenever things happen at regular intervals, such as regular meetings or train schedules, arithmetic sequences help us figure out when to expect them.

7. Population Growth: When people talk about how a city's population is growing steadily, they're using arithmetic sequences to show how many more people there are each year.

8. Art and Music: Artists and musicians use arithmetic sequences to make rhythm patterns in music or designs that keep repeating a certain way.

9. Sports and Games: When scores in games keep going up by a fixed amount, like basketball points, it's just like an arithmetic sequence. It helps us know how things are progressing.

10. Statistics and Data Analysis: When people want to make sense of numbers, like looking at how things change over time, they use arithmetic sequences to understand trends and patterns.

11. Medicine: In science, arithmetic sequences can help scientists see how things grow or change in a steady, predictable way. This is used in understanding biological processes.

12. Construction and Engineering: When builders work on things that have a regular pattern, like stairs or floors in a building, they use arithmetic sequences to plan and get everything just right.

These are just some examples of how we can use arithmetic sequences in different parts of our lives. They help us understand how things change in a predictable, step-by-step manner.

The general form of an arithmetic sequence is: $$

$a,a+d,a+2d,a+3d,\dots$

 Where:

• $\mathrm{<span\; style="color:\; \#990000;"><b>a</b></span>}$ is the first term of the sequence.
• $\mathrm{<span\; style="color:\; \#2b00fe;"><b>d</b></span>}$ is the common difference between consecutive terms.
• The terms $a+d,a+2d,a+3d,\dots$represent the subsequent terms in the sequence.

For example, consider the arithmetic sequence: $\mathrm{<span\; style="color:\; \#990000;">3</span>},7,11,15,19,\dots$

In this sequence:

• The first term $\mathrm{<span\; style="color:\; \#990000;">a</span>}$ is 3.
• The common difference $\mathrm{<span\; style="color:\; \#2b00fe;">d</span>}$ is 4 (since $7-3=4$, $11-7=4$, and so on).

Illustrations:

Given 1: Arithmetic Sequence: $3,5,7,9,11,\dots$

First term: $a$ = 3 

Common difference:$d$ = 2, (d = 5-3 = 2)

Solution:  $\mathrm{<i>add\; the </i>}$$\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><span\; style="color:\; \#cc0000;">2</span></span></span></span>}$$\mathrm{<i>\; in\; every\; term<br></i>}$

$3+\mathrm{<span\; style="color:\; \#cc0000;">2</span>}=5$ 

$5+$$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><span\; style="color:\; \#cc0000;">2</span></span></span></span>$$=7$

$\mathrm{}$$7+$$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><span\; style="color:\; \#cc0000;">2</span></span></span></span>$$=9$

$\mathrm{}\mathrm{}$$9+$$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><span\; style="color:\; \#cc0000;">2</span></span></span></span>$$=11$

$\mathrm{}$So, the terms increase by 2 in each step.

Given 2: Arithmetic Sequence: $10,7,4,1,-\mathrm{2\; <br>}$

$$First term: $a$ = 10 . 

Common difference: $d$= -3 (d = 7-10 = -3)

Solution: $\mathrm{<i>add\; the\; -3</i>}$$\mathrm{<i>\; in\; every\; term</i>}$$\mathrm{<br>}$

$10-3=7$

$7-3=4$ 

$4-3=1$

$1-3=-2$

The terms decrease by -3 in each step.

Given 3:Arithmetic Sequence: $\mathrm{<br>}$

$$First term: $a$ = -1 . 

Common difference: $d$= 0, (d = -1-(-1) = 0)

Solution:$\mathrm{<i>add\; the\; -1</i>}$$\mathrm{<i>\; in\; every\; term</i>}$$\mathrm{}$

$\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml">-1 </span></span></span>}$$\mathrm{+}-1=0$

Given 4:

$\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow></semantics></math><br>}$ Arithmetic Sequence: $2,6,10,14,18,\dots$

First term: $a$= 2 

Common difference: $d$ = 4 

Solution: 

$2+\mathrm{<span\; style="color:\; \#cc0000;">4</span>}=6$$\mathrm{<br>}$

$6+$$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><span\; style="color:\; \#cc0000;">4</span></span></span></span>$$=10$

$10+$$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><span\; style="color:\; \#cc0000;">4</span></span></span></span>$$=14$$\mathrm{<br>}$

$14+$$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><span\; style="color:\; \#cc0000;">4</span></span></span></span>$$=18$

The terms increase by 4 in each step.

Given 5:

$\mathrm{<math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow></semantics></math>}$ Arithmetic Sequence: $-7,-8,-9,-10,-11,\dots$

First term $a$:: =-7 

Common difference:$\mathrm{ \; d}$=: -1

Solution: 

$-7-1=-8$

$-8-1=-9$

$-9-1=-10$

$-10-1=-11$

The terms decrease by 1 in each step.

Let's Use the Arithmetic Sequence Formula.

The formula for the $n$th term of an arithmetic sequence is given by: $a_n=a+(n-1)d$ 

Where:

• $a$ is the first term of the sequence.
• $n$ is the position of the term in the sequence.
• $d$ is the common difference between consecutive terms.

Example 1: Given the Sequence $3,7,11,15,19,\dots$$show\; how\; the\; sequence\; work.$

First term: $\mathrm{a }$= 3 

Common difference: $\mathrm{d }$= 4

${\mathrm{<span\; style="color:\; \#2b00fe;"><b>Formula:</b></span>\; a}}_n=a+(n-1)d$ 

Solution: For the $n$th term ($a_n$):

• When $n=1$: $a_n=3+(1-1)\cdot 4=3$
• When $n=\mathrm{2:}$ $a_n=3+(2-1)\cdot 4=7$
• When $n=3$: $a_n=3+(3-1)\cdot 4=11$
• When $n=4$: $a_n=3+(4-1)\cdot 4=15$
• When $n=5$: $a_n=3+(5-1)\cdot 4=19$
• $\mathrm{}$

Now let's work on the 7th and 11th position:

• When  $n=\mathrm{<span\; style="color:\; \#cc0000;">7</span>}$:  $a_n=3+(\mathrm{<span\; style="color:\; \#cc0000;">7</span>}-1)\cdot 4=27$
• $\mathrm{}$
  

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Example 2: Given the Sequence $-2,-5,-8,-11,-14,\dots$$show\; how\; the\; sequence\; work.$

First term: $\mathrm{a }$= -2 

Common difference: $\mathrm{d }$= -3

${\mathrm{<span\; style="color:\; \#2b00fe;"><b>Formula:</b></span>\; a}}_n=a+(n-1)d$ 

Solution: For the $n$th term ($a_n$):

• When $n=1$, $a_1=-2+(1-1)\cdot (-3)=-2$
• When $n=2$, $a_2=-2+(2-1)\cdot (-3)=-5$
• When $n=3$, $a_3=-2+(3-1)\cdot (-3)=-8$
• When $n=4$, $a_4=-2+(4-1)\cdot (-3)=-11$
• When $n=5$, $a_5=-2+(5-1)\cdot (-3)=-14$

Now let's work on the 8th and 14th position:

• When  $n=8$:  $a_8=2+(8-1)\cdot$$$(−3) = −19
• $\mathrm{}$

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Example 3: Given the Sequence $<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord">1</span><span\; class="mpunct">,</span><span\; class="mspace"\; style="margin-right:\; 0.1667em;"></span><span\; class="mord">1</span><span\; class="mpunct">,</span><span\; class="mspace"\; style="margin-right:\; 0.1667em;"></span><span\; class="mord">1</span><span\; class="mpunct">,</span><span\; class="mspace"\; style="margin-right:\; 0.1667em;"></span><span\; class="mord">1</span><span\; class="mpunct">,</span><span\; class="mspace"\; style="margin-right:\; 0.1667em;"></span><span\; class="mord">1<b>,</b></span></span></span></span></span>$$\dots$$show\; how\; the\; sequence\; work.$

First term: $\mathrm{a }$= 1 

Common difference: $\mathrm{d }$= 0

${\mathrm{<span\; style="color:\; \#2b00fe;"><b>Formula:</b></span>\; a}}_n=a+(n-1)d$ 

Solution: For the $n$th term ($a_n$):

• When $n=1$, $a_1=$$\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml">1+(1-1)\cdot 0\; =0</span></span></span>}$
• When $n=2$, $a_2=$1 + (1−1)⋅0 = 0
• When $n=3$, $a_3=$1 + (1−1)⋅0 = 0
• When $n=4$, $a_4=$$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml">1\; +\; (1-1)\cdot 0=0</span></span></span>$
• When $n=5$, $a_5=$$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml">1+\; (1-1)\cdot 0=0</span></span></span>$

 Based on how it works, we can tell that all the following sequence is 1.

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Example 4: Given the Sequence $10,12,14,16,18,\dots$$show\; how\; the\; sequence\; work.$

First term: $\mathrm{a }$= 10 

Common difference: $\mathrm{d }$= 2

${\mathrm{<span\; style="color:\; \#2b00fe;"><b>Formula:</b></span>\; a}}_n=a+(n-1)d$ 

Solution: For the $n$th term ($a_n$):

• When $n=1$: $a_2=10+(1-1)\cdot 2=10$
• When $n=2$: $a_2=10+(2-1)\cdot 2=12$
• When $n=3$: $a_3=10+(3-1)\cdot 2=14$
• When $n=4$: $a_4=10+(4-1)\cdot 2=16$
• When $n=5$: $a_5=10+(5-1)\cdot 2=18$

Now you work on the 8th and 14th position:

• When  $n=8$:  $a_8=\; ?$
• $\mathrm{}$

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Example 5: Given the Sequence $-3,-2,-1,0,1,\dots$$show\; how\; the\; sequence\; work..$

First term: $\mathrm{a }$= $-3$

Common difference: $\mathrm{d }$= 1

${\mathrm{<span\; style="color:\; \#2b00fe;"><b>Formula:</b></span>\; a}}_n=a+(n-1)d$ 

Solution: For the $n$th term ($a_n$):

Solution: For the $n$th term ($a_n$):

• When $n=1$, $a_1=$ Click to answer:$$
  
• When $n=2$, $a_2=-3+(2-1)\cdot 1=-2$
• When $n=3$, $a_3=$Click to answer:$$
• When $n=4$, $a_4=-3+(4-1)\cdot 1=0$
• When $n=5$, $a_5=$ Click to answer:

These examples demonstrate how to use the formula for the $n$th term of an arithmetic sequence to find specific terms in the sequences provided.

Let's have a short practice

Arithmetic Sequence Worksheet

1. Identify the first term ($a$) and the common difference ($d$) in the following arithmetic sequences: 

   a) 6, 12, 18, 24, ... 

   b) -3, -7, -11, -15, ... 

   c) 20, 21, 22, 23, ... 

   d) 2, 5, 8, 11, ... 

   e) 100, 90, 80, 70, ...

   Click to answer:
   

2. Write the first 5 terms of an arithmetic sequence with $a=4$ and $d=-2$.

   Click to answer:
   

3. Write an arithmetic sequence where the first term ($a$) is 15 and the common difference ($d$) is 3. 

   Write the first 6 terms of the sequence.

   Click to answer:
   

4. Calculate the 10th term of the arithmetic sequence with $a=7$and $d=4$.

   Click to answer:
   

5. In an arithmetic sequence, the 7th term is 42 and the common difference is 6. 

   Click to answer:
   

   What is the first term ($a$)?

6. Find the sum of the first 12 terms of an arithmetic sequence with $a=3$ and $d=5$.

   Click to answer:
   

7. Given the arithmetic sequence $10,13,16,19,\dots$, determine the 25th term ($a_{25}$).

   Click to answer:
   

8. Create an arithmetic sequence where the first term ($a$) is -8 and the common difference ($d$) is 1. 

   Write the first 8 terms of the sequence.

   Click to answer:
   

9. The 5th term of an arithmetic sequence is 28 and the 10th term is 43. What is the common difference ($d$)?

   Click to answer:
   

10. An arithmetic sequence starts with $a=2$ and has a common difference of $d=7$. Write an expression for the $n$th term ($a_n$) of this sequence.

    Click to answer:
    

 

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