Proving a Sequence is Arithmetic: Understanding the Criteria and Key Formulas
Introduction to Arithmetic Sequences
An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is referred to as the common difference and is denoted by ( d ). For instance, consider the sequence -39, -34, -29, and so on. The difference between each pair of successive numbers is 5, indicating that this is an arithmetic sequence.Determining if a Sequence is Arithmetic
To prove whether a given sequence is arithmetic, follow these steps: Calculate the difference between each pair of successive terms. Check if all the differences are equal. For instance, consider the sequence -39, -34, -29, and so on. Calculate the difference between each pair of numbers: -34 - (-39) 5 -29 - (-34) 5 ... If all the differences are the same, then the sequence is arithmetic. In this case, since the difference between each pair of numbers is 5, the sequence is indeed arithmetic.The Explicit Formula for an Arithmetic Sequence
Once you have verified that a sequence is arithmetic, you can find the nth term using the explicit formula. The explicit formula for an arithmetic sequence is given by:a n a 1 d ? ( n - 1 )
In this formula: a n The nth term of the sequence. a 1 The first term of the sequence. d The common difference. n The position of the term in the sequence.