To effectively study arithmetic sequences, follow these steps:
1. Identify Arithmetic Sequences
First, check if the given sequence is arithmetic or not, as indicated by the provided reference. An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
- Example: 2, 4, 6, 8,... is an arithmetic sequence.
- Non-Example: 1, 3, 7, 12,... is not an arithmetic sequence.
2. Calculate the Common Difference
The next step is to calculate the common difference (d). The reference provides the formula: d=a2- a1=a3-a2=… =an-a(n-1). In simpler terms, subtract any term from the term that follows it. If the result is consistent throughout the sequence, that result is your common difference.
- Example: In the sequence 2, 4, 6, 8,...
- d = 4 - 2 = 2
- d = 6 - 4 = 2
- d = 8 - 6 = 2
- Therefore, the common difference d is 2.
3. Use Formulas to Solve Problems
After confirming that the sequence is arithmetic and finding the common difference, you can solve for the nth term or the sum of the first n terms using appropriate formulas.
nth Term Formula
The general formula for the nth term (an) of an arithmetic sequence is:
an = a1 + (n - 1)d
Where:
- an is the nth term.
- a1 is the first term.
- n is the term number (e.g., 3rd term, 10th term).
- d is the common difference.
Example: Find the 10th term of the sequence 2, 4, 6, 8,...
- a1 = 2
- n = 10
- d = 2
- a10 = 2 + (10 - 1) 2 = 2 + 9 2 = 2 + 18 = 20
- Therefore, the 10th term is 20.
Sum of n Terms Formula
The sum of the first n terms (Sn) of an arithmetic sequence is:
Sn = n/2 * (a1 + an)
Alternatively, if you don't know an:
Sn = n/2 * [2a1 + (n - 1)d]
Example: Find the sum of the first 10 terms of the sequence 2, 4, 6, 8,...
- a1 = 2
- n = 10
- d = 2
- a10 = 20 (calculated previously)
- S10 = 10/2 (2 + 20) = 5 22 = 110
- Therefore, the sum of the first 10 terms is 110.
Summary
| Step | Description |
|---|---|
| 1. Identify | Check if the sequence is arithmetic. |
| 2. Common Difference | Calculate the common difference (d). |
| 3. Use Formulas | Apply the nth term and sum formulas to solve problems related to the sequence. |
By following these steps, you can effectively study and work with arithmetic sequences.
