Arithmetic Sequence Calculator
Free Arithmetic sequence Calculator for sequences. Enter values to get step-by-step solutions with formulas and graphs.
Formula
an = a1 + (n-1)d | Sn = (n/2)(2a1 + (n-1)d)
Where an is the nth term, a1 is the first term, d is the common difference, n is the number of terms, and Sn is the sum of the first n terms. The nth term formula finds any specific term, while the sum formula calculates the total of all terms from the first through the nth.
Worked Examples
Example 1: Stadium Seating Layout
Problem: A stadium has 20 rows. The first row has 15 seats and each subsequent row has 3 more seats. Find the number of seats in the 20th row and the total seats.
Solution: a1 = 15, d = 3, n = 20\n20th row: a20 = 15 + (20-1) x 3 = 15 + 57 = 72 seats\nTotal seats: S20 = (20/2) x (15 + 72) = 10 x 87 = 870\nArithmetic mean = 870 / 20 = 43.5 seats per row
Result: 20th row: 72 seats | Total: 870 seats | Mean: 43.5 per row
Example 2: Salary with Annual Raises
Problem: An employee starts at $45,000 and receives a $2,500 raise each year. What is the salary in year 10 and total earnings over 10 years?
Solution: a1 = 45000, d = 2500, n = 10\nYear 10 salary: a10 = 45000 + (10-1) x 2500 = 45000 + 22500 = $67,500\nTotal earnings: S10 = (10/2) x (45000 + 67500) = 5 x 112500 = $562,500\nAverage salary = $562,500 / 10 = $56,250
Result: Year 10 salary: $67,500 | Total 10-year earnings: $562,500
Frequently Asked Questions
What is an arithmetic sequence and what defines it?
An arithmetic sequence (also called an arithmetic progression) is an ordered list of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by d. For example, 2, 5, 8, 11, 14 is an arithmetic sequence with first term a1 = 2 and common difference d = 3. Each term equals the previous term plus d. Arithmetic sequences appear naturally in everyday scenarios like counting by fives, monthly salary increments, or evenly spaced fence posts. They are one of the most fundamental concepts in mathematics and serve as building blocks for more advanced topics.
What is the formula for the nth term of an arithmetic sequence?
The nth term formula is: an = a1 + (n - 1) x d, where a1 is the first term, d is the common difference, and n is the position number. This formula lets you find any term without listing all preceding terms. For example, in the sequence 7, 12, 17, 22, ... the 100th term is 7 + (100 - 1) x 5 = 7 + 495 = 502. The formula can also work backwards: if you know a term value and want its position, rearrange to n = ((an - a1) / d) + 1. This is useful for determining whether a specific number belongs to a given arithmetic sequence.
How do I calculate the sum of an arithmetic sequence?
The sum of the first n terms uses the formula: Sn = (n / 2) x (2a1 + (n - 1) x d), or equivalently Sn = (n / 2) x (a1 + an) where an is the last term. This formula was famously discovered by young Carl Friedrich Gauss when asked to sum the numbers 1 through 100. He recognized that pairing the first and last terms (1 + 100 = 101, 2 + 99 = 101, etc.) creates 50 pairs of 101, giving 5,050. The formula generalizes this pairing technique to any arithmetic sequence. It is one of the most elegant and practical formulas in elementary mathematics.
How are arithmetic sequences used in real-life applications?
Arithmetic sequences model situations with constant rates of change. Linear depreciation uses them when an asset loses the same dollar amount each year: a $50,000 machine depreciating $5,000 annually follows the sequence 50000, 45000, 40000, and so on. Salary schedules with fixed annual raises form arithmetic sequences. Stacking objects in rows where each row has one more item than the previous creates arithmetic sequences. Seating arrangements in theaters (rows getting wider by a fixed number of seats), mortgage amortization with fixed principal payments, and drug dosage accumulation at regular intervals all involve arithmetic progressions.
What is the arithmetic mean and how does it relate to the sequence?
The arithmetic mean of an arithmetic sequence equals the average of the first and last terms: Mean = (a1 + an) / 2. This also equals the middle term if the number of terms is odd. For the sequence 5, 8, 11, 14, 17, the arithmetic mean is (5 + 17) / 2 = 11, which is indeed the middle (third) term. The arithmetic mean has an important property: any term in an arithmetic sequence is the arithmetic mean of its two neighbors. That is, an = (a(n-1) + a(n+1)) / 2. This property provides a quick way to verify that a sequence is arithmetic and to find missing terms within the sequence.
How do I find missing terms in an arithmetic sequence?
To find missing terms, first determine the common difference from any two known consecutive terms or from any two terms and their positions. If you know a3 = 10 and a7 = 26, then d = (26 - 10) / (7 - 3) = 4. Then use a1 = a3 - 2d = 10 - 8 = 2 to find the first term. Now fill in all missing terms: a4 = 14, a5 = 18, a6 = 22. If you have three terms and need to determine which value makes them arithmetic, use the property that the middle term equals the average of the outer two terms. This technique is commonly tested in standardized math examinations and competition problems.