Arithmetic Sequence Formula: Common Difference and nth Term

Study Notes

Arithmetic Sequences: Common Difference and Formula for nth Term

Common Difference

An arithmetic sequence is a sequence of numbers whose terms increase or decrease by a constant amount, called the common difference. For example, in the sequence 2, 5, 8, 11, each number is 3 more than the previous one. The common difference is 3.

To determine the common difference, compare the difference between consecutive terms:

In the sequence 2, 5, 8, 11, the difference between the first and second terms is 3, and the difference between the second and third terms is also 3.
In the sequence 4, 8, 12, 16, the difference between the first and second terms is 4, and the difference between the second and third terms is 4.

Formula for nth Term

The formula for the nth term of an arithmetic sequence is:

[ a_n = a_1 + (n - 1)d ]

where:

(a_n) is the nth term of the sequence
(a_1) is the first term of the sequence
(n) is the number of the term
(d) is the common difference

For example, if we want to find the 16th term of the sequence 2, 10, 18, 26, we can use the formula:

[ a_{16} = a_1 + (16 - 1)d ]

[ a_{16} = 2 + (16 - 1)(8) ]

[ a_{16} = 2 + 144 ]

[ a_{16} = 146 ]

So the 16th term of the sequence is 146.

Practical Applications

The formula for the nth term of an arithmetic sequence is useful in various mathematical and real-life contexts. For example, it can be used to find the salary of an employee after a certain number of years or to determine the interest rate of a loan after a specific period.

Conclusion

Arithmetic sequences are a fundamental concept in mathematics, and understanding the common difference and formula for the nth term is crucial for working with these sequences. Whether you're solving problems on paper or using calculators, these concepts will help you find the desired terms with ease.

Description

Learn about arithmetic sequences, including how to find the common difference and use the formula for the nth term to solve problems. This quiz covers the fundamental concepts and practical applications of arithmetic sequences.