Finding an arithmetic sequence between two numbers involves determining the terms that lie between those two numbers such that the overall sequence follows a consistent pattern of adding a constant value (the common difference).
Here's a breakdown of the process:
-
Define the Problem: You have two numbers, let's call them a and b, and you want to insert n terms between them to create an arithmetic sequence.
-
Calculate the Common Difference (d): The formula to find the common difference when inserting n terms between a and b is:
- d = (b - a) / (n + 1)
-
Generate the Arithmetic Sequence: Starting with a, add the common difference d successively to get the terms of the arithmetic sequence.
- First term: a
- Second term: a + d
- Third term: a + 2d
- ...
- (n+2)th term: a + (n+1)d = b
Example:
Let's say you want to insert 3 terms between 2 and 14 to create an arithmetic sequence.
- a = 2
- b = 14
- n = 3
Calculate the common difference:
- d = (14 - 2) / (3 + 1) = 12 / 4 = 3
Generate the sequence:
- First term: 2
- Second term: 2 + 3 = 5
- Third term: 5 + 3 = 8
- Fourth term: 8 + 3 = 11
- Fifth term: 11 + 3 = 14
The resulting arithmetic sequence is: 2, 5, 8, 11, 14.
Summary
To find an arithmetic sequence between two numbers, calculate the common difference using the formula d = (b-a)/(n+1), where 'a' and 'b' are the two numbers and 'n' is the number of terms to be inserted. Then, repeatedly add the common difference to the first number to generate the sequence.
