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Chapter 9: Problem 2

In Exercises 1鈥�6, write the first five terms of the sequence. $$ a_{n}=\left(-\frac{2}{5}\right)^{n} $$

Short Answer

Expert verified
The first five terms of the sequence are -2/5, 4/25, -8/125, 16/625, and -32/3125.

Step by step solution

01

Calculate the first term (\(a_1\))

Replace \(n\) with 1 in the given formula to find the first term. We get: \(a_1 = \left(-\frac{2}{5}\right)^1 = -\frac{2}{5}\)
02

Calculate the second term (\(a_2\))

Replace \(n\) with 2 in the formula. We get: \(a_2 = \left(-\frac{2}{5}\right)^2 = \left(\frac{2}{5}\right)^2 = \frac{4}{25}\)
03

Calculate the third term (\(a_3\))

Replace \(n\) with 3 in the formula. We get: \(a_3 = \left(-\frac{2}{5}\right)^3 = -\left(\frac{2}{5}\right)^3 = -\frac{8}{125}\)
04

Calculate the fourth term (\(a_4\))

Replace \(n\) with 4 in the formula. We get: \(a_4 = \left(-\frac{2}{5}\right)^4 = \left(\frac{2}{5}\right)^4 = \frac{16}{625}\)
05

Calculate the fifth term (\(a_5\))

Replace \(n\) with 5 in the formula. We get: \(a_5 = \left(-\frac{2}{5}\right)^5 = -\left(\frac{2}{5}\right)^5 = -\frac{32}{3125}\)

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Key Concepts

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

Geometric Sequence
A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed number. This fixed number is known as the common ratio. For the sequence given in the exercise, the formula \(a_n = \left(-\frac{2}{5}\right)^n\) helps determine the sequence, and the common ratio here is \(-\frac{2}{5}\). Each term comes from the multiplication of the preceding term by \(-\frac{2}{5}\).

Some key points about geometric sequences:
  • The common ratio is consistent for all terms.
  • They can grow or shrink, depending on whether the common ratio is greater or less than one.
  • Geometric sequences only require the first term and the common ratio to generate subsequent terms.
Understanding these features will allow you to generate terms easily from the formula.
Term Calculation
Calculating each term in a sequence involves substituting values into a formula. For geometric sequences, this process is straightforward. Let's look at how term calculations work for \(a_n = \left(-\frac{2}{5}\right)^n\).

Here's the step-by-step process:
  • Identify the term number \(n\).
  • Substitute this \(n\) into the sequence formula.
  • Solve the exponential expression for the specified term.
This approach ensures you can systematically find each term without confusion.

For example, to find the third term (\(a_3\)), replace \(n\) with 3 in the formula: \(a_3 = \left(-\frac{2}{5}\right)^3 = -\frac{8}{125}\). This pattern follows for all subsequent terms.
Exponential Expression
Exponential expressions play a crucial role in geometric sequences. In the sequence \(a_n = \left(-\frac{2}{5}\right)^n\), each term is derived from raising the base \(-\frac{2}{5}\) to the nth power. Here's how you handle exponential expressions:

Key steps involve:
  • Understand the base and exponent: The base is the constant multiplied repeatedly, while the exponent indicates how many times to multiply.
  • Maintain the negative sign: If the base is negative, it affects the sign of the term, particularly for odd exponents.
  • Calculate systematically: Raise the base to the given power to find the specific term value.
Exponential expressions ensure that sequences can be computed using simple math operations, highlighting patterns in the sequence.
Series in Mathematics
In mathematics, a series can be understood as the sum of terms of a sequence. While our focus is on sequences, knowing about series helps connect the dots.

Here's what to keep in mind:
  • A series involves adding elements of a sequence together.
  • Geometric series in particular, arise from geometric sequences.
  • The sum of terms in a geometric series can be determined using specific formulas if the series is finite.
Understanding these connections allows students to explore broader mathematical concepts beyond individual sequences. The transition from a sequence to a series broadens the scope of solving more complex problems.

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Most popular questions from this chapter

Using a Binomial Series In Exercises \(17-26,\) use the binomial series to find the Maclaurin series for the function. $$ f(x)=\frac{1}{\sqrt{4+x^{2}}} $$

Show that if $$\sum_{n=1}^{\infty} a_{n}$$ is absolutely convergent, then $$\left|\sum_{n=1}^{\infty} a_{n}\right| \leq \sum_{n=1}^{\infty}\left|a_{n}\right|$$

Finding a Taylor Series In Exercises \(1-12,\) use the definition of Taylor series to find the Taylor series, centered at \(c,\) for the function. $$ f(x)=\frac{1}{1-x}, \quad c=2 $$

Find the values of \(x\) for which the series converges. $$\sum_{n=0}^{\infty} \frac{(x+1)^{n}}{n !}$$

Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(1+\frac{2}{3}+\frac{3}{3^{2}}+\frac{4}{3^{3}}+\frac{5}{3^{4}}+\frac{6}{3^{5}}+\cdots\)
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