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Questions and Answers
What is the formula for the nth term of an arithmetic sequence?
Find the 10th term of an arithmetic sequence where the first term is 5 and the common difference is 2.
What is the arithmetic mean of 3, 5, and 7?
What is the definition of arithmetic means?
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If the 5th term of an arithmetic sequence is 11, and the common difference is 2, what is the 1st term?
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If the 3rd term of an arithmetic sequence is 7 and the 6th term is 13, what is the common difference?
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In an arithmetic sequence, if the 10th term is 25 and the 15th term is 40, what is the 20th term?
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The 2nd and 5th terms of an arithmetic sequence are 4 and 10, respectively. What is the 9th term?
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If the arithmetic mean of 3, 5, and 7 is x, what is the value of x?
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In an arithmetic sequence, if the sum of the 5th and 7th terms is 24, and the common difference is 2, what is the 10th term?
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Study Notes
Arithmetic Sequence Basics
- The nth term of an arithmetic sequence is calculated using the formula: T(n) = a + (n - 1)d, where a is the first term, d is the common difference, and n is the term number.
Finding Specific Terms
- To find the 10th term of an arithmetic sequence with the first term 5 and a common difference of 2:
- T(10) = 5 + (10 - 1) × 2 = 5 + 18 = 23*.
Arithmetic Mean
- The arithmetic mean of a set of numbers is calculated by summing the numbers and dividing by the count.
- For 3, 5, and 7, the arithmetic mean is:
- (3 + 5 + 7) / 3 = 15 / 3 = 5*.
Definition of Arithmetic Means
- Arithmetic means refer to the terms that evenly space between two given numbers in an arithmetic sequence, maintaining a constant difference.
Finding First Term from Given Terms
- If the 5th term of an arithmetic sequence is 11 and the common difference is 2, the first term can be calculated:
- T(5) = a + (5 - 1) × d = 11* leads to a + 8 = 11, thus a = 3.
Determining Common Difference
- Given that the 3rd term is 7 and the 6th term is 13, the common difference can be found:
- d = (T(6) - T(3)) / (6 - 3) = (13 - 7) / 3 = 2*.
Calculating Other Terms
- For an arithmetic sequence where the 10th term is 25 and the 15th term is 40, determine the 20th term:
The common difference can be found first as d = (40 - 25) / 5 = 3. Using T(20) = T(10) + 10d gives: - T(20) = 25 + (10 × 3) = 25 + 30 = 55*.
Finding the 9th Term
- Given the 2nd term is 4 and the 5th term is 10, use these to find the 9th term:
The common difference d = (10 - 4) / (5 - 2) = 2. Finding a, we get: - T(2) = a + (2 - 1) × d = 4* leading to a + 2 = 4, thus a = 2. Find T(9) = a + (9 - 1) × d = 2 + 16 = 18.
Sum of Specific Terms
- If the sum of the 5th and 7th terms is 24 with a common difference of 2, find the 10th term:
Using the information, let T(5) = a + 8 and T(7) = a + 12 leads to: - (a + 8) + (a + 12) = 24 → 2a + 20 = 24*, so 2a = 4, hence a = 2. Then calculate T(10) = 2 + 18 = 20.
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Description
Test your understanding of arithmetic sequences by finding the nth term, arithmetic means, and solving problems involving sequences. Apply your knowledge to a variety of questions and become proficient in this algebra topic.
