Find The Explicit Formula Calculator

Arithmetic Sequence Explicit Formula Calculator

This calculator helps you find the explicit formula, the nth term, and the sum of the first n terms of an arithmetic sequence. Enter the first term (a1), the common difference (d), and the term number (n) to get started.

Sequence Terms Table

Term (n)	Value (an)
Enter values and calculate to see the table.

Table showing the first n terms of the sequence.

Sequence Terms Chart

Chart illustrating the values of the first n terms of the arithmetic sequence.

What is an Explicit Formula Calculator for Arithmetic Sequences?

An explicit formula calculator for arithmetic sequences is a tool used to determine the value of any term in an arithmetic sequence without needing to list all the preceding terms. It also often calculates the sum of the first 'n' terms. An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Anyone studying sequences in mathematics, from middle school students to those in higher education, or professionals dealing with linear growth patterns, can use this explicit formula calculator. It simplifies finding a specific term far into the sequence or calculating the sum of a large number of terms quickly.

A common misconception is that you always need to know many terms before finding the explicit formula. In reality, you only need the first term (a1) and the common difference (d) to define the explicit formula and then use the explicit formula calculator to find any term or the sum.

Explicit Formula and Mathematical Explanation

The explicit formula for the nth term (a_n) of an arithmetic sequence is:

a_n = a₁ + (n-1)d

Where:

• a_n is the nth term
• a₁ is the first term
• n is the term number (the position of the term in the sequence)
• d is the common difference

To find the value of the nth term, we start with the first term (a₁) and add the common difference (d) a total of (n-1) times. For example, the 2nd term is a₁ + d, the 3rd term is a₁ + 2d, and so on.

The sum of the first n terms (S_n) of an arithmetic sequence is given by:

S_n = n/2 * (a₁ + a_n)

or, by substituting the explicit formula for a_n:

S_n = n/2 * (2a₁ + (n-1)d)

Our explicit formula calculator uses these formulas.

Variables Table

Variable	Meaning	Unit	Typical Range
a1	First term	Varies	Any real number
d	Common difference	Varies	Any real number
n	Term number / Number of terms	None	Positive integers (1, 2, 3, …)
an	nth term	Varies	Any real number
Sn	Sum of the first n terms	Varies	Any real number

Practical Examples (Real-World Use Cases)

Let's see how the explicit formula calculator works with some examples.

Example 1: Salary Increase

Someone starts a job with an annual salary of $50,000 (a₁ = 50000) and receives a guaranteed raise of $2,000 each year (d = 2000). What will their salary be in their 8th year (n = 8)?

Using the explicit formula calculator or the formula a_n = a₁ + (n-1)d:

a₈ = 50000 + (8-1) * 2000 = 50000 + 7 * 2000 = 50000 + 14000 = $64,000

Their salary in the 8th year will be $64,000. The calculator can also find the total earnings over 8 years.

Example 2: Savings Plan

A person saves $100 in the first month (a₁ = 100) and decides to increase their savings by $20 each subsequent month (d = 20). How much will they save in the 12th month (n = 12), and what will their total savings be after 12 months?

Using the explicit formula calculator:

a₁₂ = 100 + (12-1) * 20 = 100 + 11 * 20 = 100 + 220 = $320 (savings in the 12th month)

S₁₂ = 12/2 * (2*100 + (12-1)*20) = 6 * (200 + 220) = 6 * 420 = $2,520 (total savings after 12 months)

How to Use This Explicit Formula Calculator

1. Enter the First Term (a1): Input the starting value of your arithmetic sequence.
2. Enter the Common Difference (d): Input the constant amount added to get from one term to the next.
3. Enter the Term Number (n): Input the position of the term you wish to find, or the number of terms you want to sum. This must be a positive integer.
4. Click "Calculate" or observe results: The explicit formula calculator will update automatically or after clicking calculate, showing the explicit formula used, the value of the nth term (a_n), the value of (n-1)d, and the sum of the first n terms (S_n).
5. Review the Table and Chart: The table lists the first 'n' terms, and the chart visualizes their growth.
6. Reset: Click "Reset" to clear the fields and start over with default values.
7. Copy Results: Click "Copy Results" to copy the main outputs to your clipboard.

The results help you understand the value of a specific term and the cumulative sum up to that term in an arithmetic progression.

Key Factors That Affect Explicit Formula Results

The results from the explicit formula calculator are directly influenced by the input values:

• First Term (a1): The starting point of the sequence. A higher a1 shifts the entire sequence upwards, increasing the value of every term and the sum.
• Common Difference (d): This determines the rate of increase or decrease. A positive 'd' means the terms grow, a negative 'd' means they shrink, and d=0 means all terms are the same. A larger absolute value of 'd' results in faster change.
• Term Number (n): As 'n' increases, the value of a_n moves further from a₁ (if d is not zero), and S_n accumulates more terms, generally increasing in magnitude.
• Sign of 'd': If 'd' is positive, the sequence increases; if 'd' is negative, it decreases. This impacts whether S_n grows or potentially decreases after some point if a1 is positive and d is negative.
• Magnitude of 'n': Larger 'n' values lead to larger (n-1)d contributions, significantly affecting a_n and S_n, especially when 'd' is also large.
• Starting Point 'n=1': The formulas assume the sequence starts with n=1 as the index for the first term a1.

Frequently Asked Questions (FAQ)

What is an arithmetic sequence?

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference 'd'.

How do I find the common difference?

Subtract any term from its succeeding term (e.g., a₂ – a₁, or a₃ – a₂). If the sequence is arithmetic, this difference will be constant.

Can the common difference be negative?

Yes, if the common difference is negative, the terms of the sequence decrease.

Can the first term be zero or negative?

Yes, the first term (a1) can be any real number: positive, negative, or zero.

What if n=1?

If n=1, a₁ = a₁ + (1-1)d = a₁, and S₁ = 1/2 * (2a₁ + 0) = a₁. The 1st term is just a1, and the sum of the first term is a1.

Can I use this explicit formula calculator for a geometric sequence?

No, this calculator is specifically for arithmetic sequences, which have a common difference. Geometric sequences have a common ratio, and you'd need a geometric sequence calculator for those.

What does the explicit formula tell me?

It provides a direct way to calculate any term (a_n) in the sequence given its position 'n', the first term 'a1', and the common difference 'd', without listing all terms before it.

How is the explicit formula different from a recursive formula?

The explicit formula calculates a term directly from 'n', 'a1', and 'd'. A recursive formula defines a term based on the preceding term(s) (e.g., a_n = a_n-1 + d), requiring you to know the previous term.

Related Tools and Internal Resources

• Arithmetic Sequence Basics: Learn the fundamentals of arithmetic sequences.
• Geometric Sequence Calculator: Calculate terms and sums for geometric sequences.
• Series Summation Tool: A general tool for summing various series.
• Math Calculators: Explore other mathematical calculators.
• Sequence and Series Explained: A detailed guide on sequences and series.
• Online Math Solvers: Find tools to solve various math problems.

Using our explicit formula calculator can greatly simplify working with arithmetic sequences.