Sequence Calculator

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Reviewer: David Chen, CFA, Quantitative Analyst

This calculator provides accurate results based on standard mathematical principles for arithmetic sequences.

The **Sequence Calculator** is an essential tool for instantly determining the $n$-th term and the sum of an arithmetic progression, simplifying complex series calculations for finance, engineering, and mathematics.

Arithmetic Sequence Calculator

Detailed Calculation Steps

Arithmetic Sequence Formula:

The $n$-th term ($a_n$):

$$a_n = a_1 + (n-1)d$$

The Sum of $n$ terms ($S_n$):

$$S_n = \frac{n}{2}(2a_1 + (n-1)d)$$

Formula Source: Wikipedia – Arithmetic Progression
Alternative Source: Wolfram MathWorld

Variables:

The calculator requires three inputs to determine the properties of the sequence:

  • First Term ($a_1$): The initial value of the sequence.
  • Common Difference ($d$): The constant value added to each term to get the next term.
  • Number of Terms ($n$): The index of the final term being calculated (must be a positive whole number).

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What is an Arithmetic Sequence?

An arithmetic sequence (or arithmetic progression) is a sequence of numbers such that the difference between the consecutive terms is constant. This fixed difference is called the common difference, denoted by $d$. For example, the sequence 3, 5, 7, 9, 11, … is an arithmetic sequence with a common difference of 2.

This type of sequence is fundamental in many areas, including simple interest calculations, determining uniformly accelerating motion in physics, and various financial modeling techniques where growth is linear rather than exponential. Understanding the relationships between the first term, common difference, and number of terms is crucial for predicting future values within the series.

The ability to quickly calculate the $n$-th term ($a_n$) and the cumulative sum ($S_n$) of the terms up to that point is invaluable for rapid analysis and validation of linear growth models.

How to Calculate Arithmetic Sequence (Example):

Let’s calculate the 10th term and the sum of the first 10 terms for a sequence with $a_1=5$ and $d=3$.

  1. Identify Variables: $a_1=5$, $d=3$, $n=10$.
  2. Calculate the 10th Term ($a_{10}$): Apply the formula $a_n = a_1 + (n-1)d$. $$a_{10} = 5 + (10-1) \times 3$$ $$a_{10} = 5 + 9 \times 3 = 5 + 27 = 32$$
  3. Calculate the Sum of 10 Terms ($S_{10}$): Apply the formula $S_n = \frac{n}{2}(a_1 + a_n)$. $$S_{10} = \frac{10}{2}(5 + 32)$$ $$S_{10} = 5 \times 37 = 185$$
  4. Result: The 10th term is 32, and the sum of the first 10 terms is 185.

Frequently Asked Questions (FAQ):

  • What happens if the common difference ($d$) is negative? A negative common difference means the sequence is decreasing. The calculator handles negative differences correctly, resulting in smaller terms and a potentially smaller sum.
  • Is this the same as a Geometric Sequence? No. An arithmetic sequence uses a common *difference* (addition/subtraction), while a geometric sequence uses a common *ratio* (multiplication/division).
  • Can I calculate the sum of an infinite arithmetic sequence? For a non-zero common difference, the sum of an infinite arithmetic sequence diverges (approaches positive or negative infinity). This calculator focuses on finite sums.
  • What is the minimum number of terms ($n$) I can input? The number of terms ($n$) must be a positive integer, meaning the minimum valid input is 1. Inputting 0 or a negative number will result in an error.
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