Finding the common difference in an arithmetic sequence (AP) when you only know the first and last terms requires using the formula for the nth term of an arithmetic sequence. This formula connects the first term, the common difference, the number of terms, and the last term.
Understanding Arithmetic Sequences
An arithmetic sequence is a series of numbers where the difference between consecutive terms remains constant. This constant difference is called the common difference (often denoted as 'd').
The Formula
The general formula for the nth term of an arithmetic sequence is:
an = a1 + (n - 1)d
Where:
- an is the nth term (the last term in our case)
- a1 is the first term
- n is the number of terms
- d is the common difference
Finding the Common Difference
To find the common difference (d), we can rearrange the formula:
d = (an - a1) / (n - 1)
This formula directly calculates 'd' using the first term (a1), the last term (an), and the total number of terms (n). Crucially, you need to know the number of terms (n) in the sequence to use this formula.
Example:
Let's say the first term (a1) is 5, the last term (an) is 45, and there are 16 terms (n = 16). Using the formula above:
d = (45 - 5) / (16 - 1) = 40 / 15 = 8/3
Therefore, the common difference is 8/3.
Alternative Approach (If 'n' is unknown):
If the number of terms ('n') is unknown, finding the common difference directly from only the first and last terms is impossible. You would need additional information about the sequence, such as the sum of the series or another term within the sequence.
Common Mistakes to Avoid
- Confusing Arithmetic and Geometric Sequences: Remember this method applies only to arithmetic sequences, not geometric sequences (where terms are multiplied by a constant ratio).
- Incorrect Formula Application: Double-check that you are using the correct formula and substituting the values correctly.
