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Questions and Answers
What is the formula for the nth term of an arithmetic sequence?
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.
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 arithmetic mean of 3, 5, and 7?
What is the definition of arithmetic means?
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?
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?
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?
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?
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?
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?
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.
