Questions and Answers
In an arithmetic sequence where the first term is 12 and the fifth term is 24, what is the common difference?
3
Calculate the 15th term of an arithmetic sequence where $a_1 = 4$ and the common difference is $-2$.
-26
If the first term of an arithmetic sequence is 5 and the common difference is 3, what is the 10th term?
32
If an arithmetic sequence has a first term of 7 and a seventh term of 35, what is the value of the common difference?
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An arithmetic sequence has its first term as $-10$ and its common difference as $5$. Determine the 20th term.
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Study Notes
Arithmetic Sequences Basics
- An arithmetic sequence is defined by a first term and a common difference, with each term computed by adding the common difference to the previous term.
- The nth term of an arithmetic sequence can be calculated using the formula:
- ( a_n = a_1 + (n-1)d )
- Where ( a_n ) is the nth term, ( a_1 ) is the first term, ( d ) is the common difference, and ( n ) is the position of the term.
Example Calculations
-
10th Term Calculation
- First term: 5
- Common difference: 3
- 10th term:
- ( a_{10} = 5 + (10-1) \times 3 = 5 + 27 = 32 )
-
Common Difference Calculation
- First term: 12
- Fifth term: 24
- Common difference:
- Since the fifth term can be expressed as ( a_5 = a_1 + (5-1)d ):
- ( 24 = 12 + 4d )
- Rearranging gives ( 4d = 12 ), thus ( d = 3 )
-
15th Term Calculation with Negative Difference
- First term: 4
- Common difference: -2
- 15th term:
- ( a_{15} = 4 + (15-1)(-2) = 4 - 28 = -24 )
-
Finding Common Difference from Terms
- First term: 7
- Seventh term: 35
- Expression setup:
- ( a_7 = a_1 + (7-1)d ) translates to ( 35 = 7 + 6d )
- Solving yields ( 6d = 28 ), resulting in ( d = \frac{14}{3} ) or approximately 4.67
-
20th Term Calculation
- First term: -10
- Common difference: 5
- 20th term:
- ( a_{20} = -10 + (20-1)5 = -10 + 95 = 85 )
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Description
Test your knowledge of arithmetic sequences with this quiz! You will solve problems involving the first term, common difference, and specific term calculations. Challenge yourself with multiple questions and enhance your understanding of this fundamental math concept.
