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Chapter 10: Problem 11

Evaluate each geometric sum. $$\sum_{k=0}^{20}\left(\frac{2}{5}\right)^{2 k}$$

Short Answer

Expert verified
Based on the given step-by-step solution, evaluate the geometric sum, $$\sum_{k=0}^{20}\left(\frac{2}{5}\right)^{2k}$$. The sum is as follows: $$S_{21} = \frac{25-25\left(\frac{2}{5}\right)^{42}}{21}$$

Step by step solution

01

Identify the values of n, a_1, and r

First, we need to identify the values of the variables n, a_1, and r from the given sum: $$\sum_{k=0}^{20}\left(\frac{2}{5}\right)^{2 k}$$ Here, we have - n = 21 (note that there are 21 terms as we are considering the terms from 0 to 20 inclusive) - \(a_1 = \left(\frac{2}{5}\right)^{2\cdot0} = (\frac{2}{5})^0 = 1\) - r = \(\left(\frac{2}{5}\right)^{2}\) Now that we have identified the values, we can proceed to compute the sum.
02

Use the formula for the sum of a geometric series

Using the formula for the sum of a geometric series, we compute the sum as follows: $$S_n = \frac{a_1(1-r^n)}{1-r}$$ Substitute the values of n, a_1, and r into the formula: $$S_{21} = \frac{1\left(1-\left(\frac{2}{5}\right)^{2\cdot21}\right)}{1-\left(\frac{2}{5}\right)^{2}}$$
03

Simplify the expression

Now, we will simplify the expression and find the value of the sum: $$S_{21} = \frac{1\left(1-\left(\frac{2}{5}\right)^{42}\right)}{1-\left(\frac{4}{25}\right)}$$ $$S_{21} = \frac{1-\left(\frac{2}{5}\right)^{42}}{\frac{21}{25}}$$ Multiply both numerator and denominator by 25 to eliminate fractions within fractions: $$S_{21} = \frac{25-25\left(\frac{2}{5}\right)^{42}}{21}$$ Now, we have the expression for the sum: $$S_{21} = \frac{25-25\left(\frac{2}{5}\right)^{42}}{21}$$ By evaluating the sum using this expression, we can find the answer to the exercise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of a Geometric Series
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. When you want to calculate the total of a geometric series, you can use a handy formula. This formula is particularly useful when the series has a finite number of terms. The formula for the sum of the first \( n \) terms of a geometric series is:
  • \( S_n = \frac{a_1(1-r^n)}{1-r} \)
where:
  • \( S_n \) represents the sum of the first \( n \) terms.
  • \( a_1 \) denotes the first term.
  • \( r \) is the common ratio.
If the ratio \( |r| < 1 \), the terms get progressively smaller, making the formula straightforward to use for evaluating sums. In some cases, even with a small ratio, a large exponent like \( r^n \) tends toward zero, simplifying calculations even further.
Geometric Progression
A geometric progression is a sequence where each term is derived by multiplying the previous term by a constant called the common ratio. This sequence forms the heart of a geometric series.
  • The formula for the general term of a geometric progression is \( a_k = a_1 \, r^{k-1} \),
where \( a_k \) is the term you want to find.For instance, in the given exercise, we can identify this sequence formed by the terms \( (\frac{2}{5})^{2k} \). This sequence grows exponentially or declines based on the value of \( r \), illustrating the beauty and complexity of geometric progressions.When analyzing such sequences, focus on:
  • The initial term (\( a_1 \)).
  • The common ratio (\( r \)).
  • How these affect the behavior of the sequence.
These sequences are versatile and appear in many mathematical and real-world applications, such as in calculating interest and population growth models.
Series Evaluation
Evaluating a series involves finding the sum or total value resulting from all the terms in the series. For a geometric series, evaluating it efficiently requires you to apply its specific summation formula.To evaluate the geometric series in the exercise, follow these steps:
  • Identify the number of terms \( n \), the first term \( a_1 \), and the common ratio \( r \).
  • Substitute these values into the formula for the sum of a geometric series.
  • Perform algebraic simplifications, if necessary, to reach the solution.
In our specific case, where the terms include powers like \( (\frac{2}{5})^{2k} \), ensure precision in handling these elevated powers during calculations.Through evaluating geometric series, you not only solve mathematical problems but also gain insights into theoretical and practical implications, like understanding growth patterns or depreciation in financial scenarios.

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Most popular questions from this chapter

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty} \frac{1}{(k+1) !-k !}$$

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{\sqrt{\ln (k+2)}}$$

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty} \frac{(-1)^{k} k}{k^{3}+1}$$

Determine whether the following statements are true and give an explanation or counterexample. a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series. b. The series \(\sum_{k=3}^{\infty} \frac{1}{k \ln ^{p} k}\) converges for the same values of \(p\) as the series \(\sum_{k=3}^{\infty} \frac{1}{k^{p}}\) c. Both the Ratio Test and the Root Test can be applied conclusively to a geometric series. d. The Alternating Series Test can be used to show that some series diverge.

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{1}{n}=0$$
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