ARITHMETIC SEQUENCE MADE EASY

 ARITHMETIC SEQUENCE

A natural sequence typically refers to the sequence of natural numbers, which includes all positive integers starting from 1. This sequence can be expressed as:

$$\mathbf{<br>}$$

Properties of Natural Sequences

1. Ordering. The natural sequence is totally ordered, meaning each number has a unique position.
2. Successor Function. Each natural number a has a successor S(a)=a+1, which is a fundamental concept in the Peano axioms that define natural numbers.
3. Induction. Any subset of natural numbers that contains 1 and is closed under the successor function must include all natural numbers.

Patterns in Natural Sequences

Natural sequences can exhibit various patterns, including:

1. Arithmetic Patterns:
   • In an arithmetic sequence, each term is generated by adding a constant (the common difference) to the previous term. For example, in the sequence $1,2,3,4,\dots \; the\; common\; difference\; is$$1$.
2. Geometric Patterns:
   • A geometric sequence involves multiplying by a constant factor. For example, $1,2,4,8,\dots$, is a geometric sequence where each term is multiplied by $\mathrm{<br>}$.
3. Fibonacci Sequence:
   • This famous sequence starts with $\mathrm{<br>}$ and $1$, and each subsequent term is the sum of the two preceding ones: $0,1,1,2,3,5,8,\dots$
4. Triangular Numbers:
   • The triangular number sequence represents counts of objects arranged in an equilateral triangle. The sequence begins as $1,3,6,10,\dots ,\; where\; each\; term\; can\; be\; found\; using\; the\; formula$$S_n=\frac{n(n+1)}{2}$.
5. Square Numbers:
• This pattern includes numbers that are squares of integers: $1^2=1,2^2=4,3^2=9,\dots$ resulting in the sequence $1,4,9,\mathrm{16,}$.

An arithmetic sequence (or arithmetic progression) is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference and can be positive, negative, or zero.

Definition

An arithmetic sequence can be defined as follows:

• If $a_1$ is the first term and $d$ is the common difference, then the sequence can be expressed as:
  $$a_1,a_1+d,a_1+2d,a_1+3d,\dots$$
  
• For example, in the sequence $2,5,8,11,\dots ,\; the\; first\; term$${\mathrm{<span\; style="color:\; \#990000;">a</span>}}_{\mathrm{<span\; style="color:\; \#990000;">1</span>}}<span\; style="color:\; \#990000;">=</span>\mathrm{<span\; style="color:\; \#990000;">2</span>}$ and the common difference $d=3$.

Formulae

The nth Term

The formula to find the n-th term ($a_n$) of an arithmetic sequence is:

where:

• ${\mathrm{<span\; style="color:\; \#cc0000;">a</span>}}_{\mathrm{<span\; style="color:\; \#cc0000;">n</span>}}$ is the n-th term,
• ${\mathrm{<span\; style="color:\; \#cc0000;">a</span>}}_{\mathrm{<span\; style="color:\; \#cc0000;">1</span>}}$ is the first term,
• $\mathrm{<span\; style="color:\; \#cc0000;">d</span>}$ is the common difference,
• $\mathrm{<span\; style="color:\; \#cc0000;">n</span>}$ is the term number.

The Sum of First n Terms

The sum ($S_n$) of the first n terms of an arithmetic sequence can be calculated using:

or alternatively,

Examples

1. Example 1: Starting at 2 with a common difference of 3: 
                          Sequence: 2, 5, 8, 11, 14, ...

2. Example 2: Starting at 10 with a common difference of -2: 
                          Sequence: 10, 8, 6, 4, 2, ...

3. Example 3: Starting at 0 with a common difference of 1: 
                          Sequence: 0, 1, 2, 3, 4, ...

4. Example 4: Starting at 5 with a common difference of 5: 
                          Sequence: 5, 10, 15, 20, 25, ...

5. Example 5: Starting at 100 with a common difference of -10: 
                          Sequence: 100, 90, 80, 70, 60, ...

These examples illustrate the concept of arithmetic sequences with varying starting points and common differences.

Supplemental Understanding of Arithmetic Sequences

An arithmetic sequence (or arithmetic progression) is one of the most fundamental concepts in algebra. It appears in various real-world applications, from calculating loan payments to predicting patterns in numbers.

In this guide, we’ll cover:
✔ What is an arithmetic sequence?
✔ The explicit and recursive formulas
✔ Step-by-step examples (easy to hard)
✔ Practice problems with solutions

Let’s make it!

1. What is an Arithmetic Sequence?
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the common difference (d).

Example of an Arithmetic Sequence:

Consider the sequence:
3, 7, 11, 15, 19, ...

> The common difference (d) is 7−3 = 4.

> Each term increases by 4.

2. Arithmetic Sequence Formulas

A. Recursive Formula

The next term is based on the previous term:

where;

B. Explicit Formula (General Term)

Finds any term directly without knowing the previous on

where;

3. Examples (Easy to Hard)

Example 1: Finding the Common Difference

Given sequence: 5,9,13,17,21,…

Find the common difference (d) and the next term.

Solution: Solve the d first.

Example 2: Finding the nth Term

Solution:

Using the explicit formula:

Example 3: Finding Missing Terms

Find the first term (a1​) and common difference (d).

Solution:
Set up two equations using the explicit formula:

Subtract the first equation from the second:

Now, substitute d = 4 into the first equation:

Example 4: Word Problem (Hard)

Problem: A taxi charges a base fare of $5 and $2 per kilometer. Write an arithmetic sequence for the total fare after 1 km, 2 km, 3 km, etc. Then, find the fare after 15 km.

Solution:

Using explicit formula;

For n = 15

Answer: The fare after 15 km is $35.

4. Practice Problems (With Answers)

Test your understanding with these problems:

Simple exercises.

Arithmetic Sequence Exercises


1. Find the 10th term of the arithmetic sequence where the first term a1= 5 and the common difference d=3.


2. Given an arithmetic sequence with a1=10 and d=−2d, find the 15th term.


3. The 5th term of an arithmetic sequence is 17, and the common difference is 4. Find the 1st term.


4. If the 3rd term of an arithmetic sequence is 12 and the 7th term is 28, find the common difference d.


5. Write the first 5 terms of an arithmetic sequence where a1=7 and d=−1.


Arithmetic Series Exercises


6. Find the sum of the first 20 terms of the arithmetic sequence where a1=3 and d=2.


7. Calculate the sum of the first 15 terms of the arithmetic sequence: 4, 7, 10, 13, ...


8. If the sum of the first 10 terms of an arithmetic sequence is 120 and the first term is 5, find the common difference d.


9.  Find the 5th term of the sum of the first n terms of an arithmetic sequence is given by 

.

10. An arithmetic series has .  Find the number of terms n.

Real-Life Scenario: Saving Money with Arithmetic Sequences!

Situation:
Liam wants to save money for a new gaming console that costs $600. He decides to save in a special way:

Week 1: He saves $20.

Each following week: He saves $5 more than the previous week.

Problem:

Will he have enough money after 10 weeks?
🔍 Solution Using Arithmetic Sequence & Series

1. Identify the sequence:

2. First term (a1) = $20 (Week 1)

3. Common difference (d) = $5 (increase per week)

Find Week 10’s savings (a10):

Calculate TOTAL savings in 10 weeks (S10​):

Using sum formula:

So, Liam only has **425∗∗after10weeks

💡 How Arithmetic Sequences Helped:


✅ Predicted savings growth over time.
✅ Calculated exact totals to plan better.
✅ Realized he needs to adjust (save longer or increase weekly amounts).


Liam’s Next Move:

He extends his plan to 15 weeks and recalculates—this time, he succeeds!


Moral: Arithmetic sequences aren’t just math—they’re life skills for smart planning! 

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