Arithmetic Sequence Worksheet
An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.
Here are some examples of arithmetic sequences:
- 2, 4, 6, 8, 10... (common difference = 2)
- 10, 7, 4, 1, -2... (common difference = -3)
- -5, -1, 3, 7, 11... (common difference = 4)
Key Formulas for Arithmetic Sequences:
- nth term (a<sub>n</sub>): a<sub>n</sub> = a<sub>1</sub> + (n-1)d
- a<sub>1</sub> is the first term
- d is the common difference
- n is the number of the term in the sequence
- Sum of the first n terms (S<sub>n</sub>): S<sub>n</sub> = n/2 [2a<sub>1</sub> + (n-1)d]
Here are some practice problems you can try:
Problem 1:
Find the 10th term of the arithmetic sequence: 3, 7, 11, 15...
Solution:
- Identify the first term (a<sub>1</sub>) and the common difference (d).
- a<sub>1</sub> = 3
- d = 7 - 3 = 4
- Use the formula for the nth term:
- a<sub>10</sub> = 3 + (10 - 1)4 = 3 + 36 = 39
Therefore, the 10th term of the arithmetic sequence is 39.
Problem 2:
Find the sum of the first 20 terms of the arithmetic sequence: -2, 1, 4, 7...
Solution:
- Identify the first term (a<sub>1</sub>) and the common difference (d).
- a<sub>1</sub> = -2
- d = 1 - (-2) = 3
- Use the formula for the sum of the first n terms:
- S<sub>20</sub> = 20/2 [2(-2) + (20 - 1)3] = 10 (-4 + 57) = 530
Therefore, the sum of the first 20 terms of the arithmetic sequence is 530.
Problem 3:
The 5th term of an arithmetic sequence is 12 and the 10th term is 27. Find the first term and the common difference.
Solution:
- Use the formula for the nth term to set up two equations:
- a<sub>5</sub> = a<sub>1</sub> + 4d = 12
- a<sub>10</sub> = a<sub>1</sub> + 9d = 27
- Solve this system of equations to find a<sub>1</sub> and d.
- Subtracting the first equation from the second equation, we get:
- 5d = 15
- d = 3
- Substitute d = 3 into either equation to find a<sub>1</sub>:
- a<sub>1</sub> + 4(3) = 12
- a<sub>1</sub> = 0
- Subtracting the first equation from the second equation, we get:
Therefore, the first term of the arithmetic sequence is 0 and the common difference is 3.
This worksheet is just a starting point. You can find many more practice problems and resources online. Remember to focus on understanding the concepts of arithmetic sequences and the formulas used to solve problems. Good luck!
