Math Problem Statement
Solution
Let's break down and solve both questions from the image:
Question 7:
Given the sequence , we are asked to find the sum of the first 100 terms, .
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Recognize the Sequence Type: This sequence appears to be arithmetic because the difference between consecutive terms is constant. Therefore, the common difference .
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Formula for the Sum of an Arithmetic Sequence: The sum of the first terms of an arithmetic sequence is given by: where:
- is the first term,
- is the common difference,
- is the number of terms.
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Substitute Known Values:
- First term ,
- Common difference ,
- Number of terms .
Substituting these into the formula: The correct answer for Question 7 is C. 37,850.
Question 8:
Given the sequence , we are asked to find the sum of the first 100 terms, .
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Recognize the Sequence Type: This is again an arithmetic sequence with a constant difference. Therefore, the common difference .
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Sum of the First 100 Terms: Using the same formula for the sum:
- First term ,
- Common difference ,
- Number of terms .
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Substitute the Values: The correct answer for Question 8 is B. -30,050.
Do you have any questions, or would you like more details on any of the steps?
Here are 5 related questions you might find useful:
- How do you identify if a sequence is arithmetic?
- Can you explain how to derive the formula for the sum of an arithmetic series?
- What is the difference between an arithmetic and a geometric sequence?
- How would you find the 100th term of an arithmetic sequence?
- Can an arithmetic sequence have a common difference of 0?
Tip: When working with arithmetic sequences, always check for a constant difference between consecutive terms to confirm the sequence type.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequences
Summation of Arithmetic Series
Formulas
Sum of an arithmetic sequence: S_n = (n / 2) * (2a + (n - 1) * d)
Theorems
Arithmetic sequence summation
Suitable Grade Level
Grades 9-12
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