How to Figure Out if a Number is in an Arithmetic Sequence?

To determine if a number is part of an arithmetic sequence, you first need to examine if the sequence itself is arithmetic. Let's break down the process:

Defining an Arithmetic Sequence

An arithmetic sequence is a series of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference.

How to Determine If a Sequence is Arithmetic

To check if a sequence is arithmetic, you need to calculate the difference between several pairs of consecutive terms. Here’s the process:

1. Identify consecutive terms: Select pairs of numbers that follow each other directly in the sequence.

2. Calculate the differences: Subtract the first term in each pair from the second term.

3. Check for a common difference: If the differences you calculated in step 2 are all the same, then the sequence is arithmetic. According to the provided reference, "If the sequence has a common difference, it is arithmetic."

   • If the differences are not the same, then the sequence is not arithmetic.

Example

Let's test this with a sequence: 2, 5, 8, 11, 14

Consecutive Terms	Difference
5 - 2	3
8 - 5	3
11 - 8	3
14 - 11	3

Since the difference is consistently 3, this is an arithmetic sequence.

Checking If a Number Is in an Arithmetic Sequence

Now, you have determined a sequence is arithmetic. You want to figure out if a specific number is within it.

1. Know the first term and common difference: You need these two values from the sequence. Let’s call them a (the first term) and d (the common difference).

2. Use the formula: The general formula for the nth term of an arithmetic sequence is:

   an = a + (n - 1)d

   Where:

   • an is the nth term in the sequence
   • a is the first term
   • n is the position of the term in the sequence
   • d is the common difference.

3. Test the number:

   • Substitute an with the number you want to check.
   • Use the known first term (a) and the common difference (d).
   • Solve for n. If n is a positive integer, the tested number is part of the sequence. If n is not a positive integer (i.e., it's a fraction, decimal or negative), the tested number is not in the sequence.

Example

Let's use the arithmetic sequence from before (2, 5, 8, 11, 14). We know a=2 and d=3. Is the number 20 in this sequence?

1. Substitute the values: 20 = 2 + (n - 1)3
2. Solve for n:
   • 20 = 2 + 3n - 3
   • 20 = 3n - 1
   • 21 = 3n
   • n = 7

Because n is a positive integer (7), then 20 is in this arithmetic sequence and would be the 7th term.

What about the number 15?

1. Substitute: 15 = 2 + (n-1)3
2. Solve for n:
   • 15 = 2 + 3n - 3
   • 15 = 3n - 1
   • 16 = 3n
   • n = 16/3 which is not a whole number

Because n is not a positive integer, then 15 is not in this arithmetic sequence.

Conclusion

In short, to know if a number is in an arithmetic sequence, you first confirm the sequence is arithmetic by looking for a common difference between consecutive terms. Then, apply the formula for the nth term, using the specific number you want to check as the an and determine if there's a whole number for n. If so, the number is in the sequence.