Questions and Answers
What is the common difference of the arithmetic sequence on which the series 3 + 4.5 + 6 + 7.5 + 9 + 10.5 is based?
1.5
What is the explicit rule for the sequence based on the series?
a_k = 3 + (k-1) * 1.5
How is the arithmetic series written using summation notation?
A = (1.5 + 1.5k)
What are the terms of the arithmetic series S_4 = sum k=1 ^ 4 (-3 + 9k)?
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What is the common difference of the arithmetic sequence on which the series S_4 is based?
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What is the value of the arithmetic series S_4?
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What is the 30th term of the arithmetic sequence -20, -16, -12, -8, -4,...?
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Which expression represents the sum of the first 30 terms of the sequence?
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What is the sum of the first 30 terms of the sequence?
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What is the explicit rule for the arithmetic sequence based on the series 12+18+24+30+...+198?
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How many terms are in the series 12+18+24+30+...+198?
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What is the value of the arithmetic series 12+18+24+30+...+198?
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What arithmetic series represents the total distance Miguel has run after he has trained n times?
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What is the least number of times Miguel must train to complete a marathon of 26.2 miles?
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What is the 100th even natural number?
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What is the sum of the first 100 even natural numbers?
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Complete the sentence: The sum of the first 100 even natural numbers is [blank] the sum of the first 101 odd natural numbers.
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What is the least score in a game played with a deck of cards?
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What is another score recorded in the card game?
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What is the highest score recorded in a card game?
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Study Notes
Arithmetic Sequences and Series
- An arithmetic sequence is characterized by a constant common difference between consecutive terms.
- For the series 3 + 4.5 + 6 + 7.5 + 9 + 10.5, the common difference is 1.5.
- The explicit rule for the sequence is given by ak = 3 + (k - 1) * 1.5.
- Summation notation for this arithmetic series is represented as S = (1.5 + 1.5k).
Evaluating Arithmetic Series
- For the series defined by S4 = sum k=1^4 (-3 + 9k), the terms are A1 = 6, A2 = 15, A3 = 24, A4 = 33.
- The common difference of this series is 9.
- The total value of the arithmetic series S4 is calculated to be 78.
Specific Term Calculation
- In the sequence -20, -16, -12, -8, -4, the 30th term (a30) equals 96.
- The sum of the first 30 terms in this sequence is given as S30 = 1140.
Additional Arithmetic Series
- For the series 12 + 18 + 24 + 30 + ... + 198, the explicit rule is identified as starting at 12 with a common difference of 6.
- The total number of terms in this series is 32.
- The value of the entire arithmetic series is found to be 3360.
Real-life Application
- Miguel runs distances starting at 0.5 miles, increasing the distance by 0.2 miles with each training session. His total distance after n training sessions is modeled by the arithmetic series C = (0.3 + 0.2k).
- In terms of distance training for a marathon of 26.2 miles, Miguel must train a minimum of 15 times to prepare adequately.
Even and Odd Number Series
- The 100th even natural number is determined to be 200.
- The sum of the first 100 even natural numbers equals 10100.
- The sum of the first 100 even natural numbers is less than the sum of the first 101 odd natural numbers.
Scoring in a Card Game
- A game involving a deck of cards can yield a minimum score of 28 points, with a maximum potential being 329 points.
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Test your knowledge of arithmetic sequences and series with these Algebra 2 flashcards. Challenge yourself with questions on common differences, explicit rules, and summation notation. Perfect for students looking to reinforce their understanding of the topic!
