{Use of Tech} For what value of r does
∑ (k = 3 to ∞) r²ᵏ = 10?

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Recognize that the given series is an infinite geometric series starting from k = 3, with the general term \( r^{2k} \). This means the terms are \( r^{6}, r^{8}, r^{10}, \ldots \).

Identify the first term \( a \) of the series as \( r^{6} \) (when \( k=3 \)) and the common ratio \( q \) as \( r^{2} \) because each term increases the exponent by 2.

Recall the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - q} \), which converges if \( |q| < 1 \). Here, the sum is given as 10, so set up the equation \( \frac{r^{6}}{1 - r^{2}} = 10 \).

Multiply both sides of the equation by \( 1 - r^{2} \) to clear the denominator, resulting in \( r^{6} = 10(1 - r^{2}) \).

Rewrite the equation to isolate terms and form a polynomial in terms of \( r \), which can then be solved to find the value(s) of \( r \) that satisfy the equation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio r. The series ∑ r^k from k = n to ∞ converges if |r| < 1, and its sum can be calculated using the formula S = a / (1 - r), where a is the first term.

Sum of an Infinite Geometric Series Starting at k = n

When a geometric series starts at an index k = n instead of 0, the first term a is r^n. The sum from k = n to ∞ is S = r^n / (1 - r), assuming |r| < 1. This adjustment is crucial for correctly evaluating series that do not start at zero.

Solving for the Common Ratio in an Infinite Series

To find the value of r that satisfies a given sum, set the sum formula equal to the target value and solve the resulting equation. This often involves algebraic manipulation and considering the convergence condition |r| < 1 to ensure the series sum is finite.

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