The explicit formula for an arithmetic progression is a way to directly calculate any term in the sequence without needing to know the preceding terms.
Understanding the Explicit Formula
According to the provided video reference, an explicit formula allows you to jump directly to any term in the sequence. The video shows that the explicit formula can be expressed as:
an = a1 + (n - 1)d
Where:
- an represents the nth term in the sequence.
- a1 is the first term of the sequence.
- n is the position of the term in the sequence (e.g., 1st, 2nd, 3rd, etc.).
- d is the common difference between consecutive terms.
How to Use the Explicit Formula
To use the formula, you need to know the first term (a1) and the common difference (d). Then, if you are asked to find the 20th term (a20), you can substitute n=20 into the formula.
Example
Let's consider an arithmetic sequence where the first term is 3 (a1 = 3) and the common difference is 2 (d = 2).
- To find the 5th term (a5):
- a5 = 3 + (5 - 1) * 2
- a5 = 3 + (4) * 2
- a5 = 3 + 8
- a5 = 11
Therefore, the 5th term in this arithmetic sequence is 11.
Advantages of the Explicit Formula
- Direct Calculation: It allows you to calculate any term directly without knowing the previous terms.
- Efficiency: It's particularly useful when you need to find a term far down the sequence.
- Generalization: It provides a formula that applies to all arithmetic sequences.
In summary, the explicit formula provides a shortcut to find any term in an arithmetic sequence, making calculations much faster and easier.
