Math Problem Statement
Solution
We are tasked with finding the sum of an infinite geometric sequence. Let us break it down step by step.
Problem Breakdown:
-
Given:
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Geometric sequence relationship:
- Each term is related by the common ratio :
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Find the sum of the infinite geometric series:
- The formula for the sum of an infinite geometric sequence is:
Step 1: Find the common ratio
From the given terms and : Substituting and : Solve for :
Step 2: Find
We know . Substituting and :
Step 3: Find the sum of the infinite geometric sequence
Using the sum formula: substitute and :
Simplify the denominator :
The sum becomes:
Rationalize the denominator: Simplify further:
Final Answer:
Would you like further clarification or details? Here are some related questions:
- What are the general properties of geometric sequences?
- How is the formula for the sum of an infinite series derived?
- How does the value of affect the convergence of the series?
- Can this method be extended to find partial sums of the sequence?
- What is the impact of rationalizing the denominator in mathematical computations?
Tip: Always ensure before attempting to compute the sum of an infinite geometric series.
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Math Problem Analysis
Mathematical Concepts
Sequences and Series
Geometric Sequences
Infinite Series
Common Ratio
Formulas
aₙ₊₁ = r * aₙ (geometric sequence relation)
Sum of infinite geometric series: S = a₁ / (1 - r)
Theorems
Convergence of geometric series (if |r| < 1)
Suitable Grade Level
Grades 10-12
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