Math Problem Statement
Solution
Let's go through the questions in the image step by step.
1. Find the specified sum for each of the following geometric series:
a. find
This is a geometric series with:
- First term
- Common ratio
The sum of the first terms of a geometric series is given by the formula: For , we plug in:
b. find
This is another geometric series with alternating signs:
- First term
- Common ratio
Again, using the formula for the sum of the first terms:
c. find
This geometric series has:
- First term
- Common ratio
Applying the sum formula:
2. For the geometric sequence write an expression for:
a. The -th term
In a geometric sequence, the -th term is given by: Here, and . So the expression for the -th term is:
b. The sum of the first terms
Using the sum formula again: For this series, and , so:
3. For the geometric sequence , write an expression for:
a. The -th term
Here:
- First term
- Common ratio
So the -th term is: Simplifying:
b. The sum of the first terms
Using the sum formula for geometric series: Simplifying:
Would you like a detailed breakdown of any particular solution or calculation?
Here are 5 relative questions for further practice:
- How would the sum formula change if the common ratio is negative?
- Can you find for the sequence ?
- For the series , calculate the sum of the first 7 terms.
- How would you calculate the 12th term of the series ?
- What happens to the sum of an infinite geometric series if ?
Tip: When working with geometric series, always check if the common ratio is between -1 and 1, as this allows for the sum of an infinite series to exist.
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Math Problem Analysis
Mathematical Concepts
Geometric Series
Sequences
Summation
Formulas
Sum of the first n terms of a geometric series: S_n = a(1 - r^n) / (1 - r)
n-th term of a geometric sequence: a_n = a * r^(n-1)
Theorems
Geometric series summation
Suitable Grade Level
Grades 9-12