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Chapter 9: Problem 7

Find a formula for \(s_{n-1} n \geq 1\) $$4,8,16,32,64, \dots$$

Short Answer

Expert verified
The formula is \( s_{n-1} = 2^{n+1} \).

Step by step solution

01

Analyze the Sequence

The given sequence is 4, 8, 16, 32, 64, ... We need to identify the pattern or rule for generating these terms. Each number is double the previous number.
02

Express the Pattern Mathematically

Let's express the sequence in terms of powers of 2. Observing the given terms:- 4 can be written as \(2^2\)- 8 can be written as \(2^3\)- 16 can be written as \(2^4\)- 32 can be written as \(2^5\)- 64 can be written as \(2^6\)This suggests a pattern where the nth term is \(2^{n+1}\).
03

Generalize the Formula

To find a formula for the \(n\)-th term, observe that the sequence starts at \(n=1\) with the value 4, which matches \(2^{2}\) . Therefore, the \(n\)-th term, which is the \(s_{n-1}\), can be represented by the formula:\[ s_{n-1} = 2^{n+1} \]

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Key Concepts

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

Patterns in Sequences
Sequences in mathematics are sets of numbers arranged in a specific order based on some rule or pattern. Identifying patterns is the first step in understanding sequences.
For the sequence 4, 8, 16, 32, 64, it's apparent that each term doubles the previous one.
This pattern is described as a geometric sequence.
  • In geometric sequences, each term is a result of multiplying the previous term by a fixed, non-zero number called the common ratio.
  • Here, the common ratio is 2, since each term is 2 times the term before it.
Recognizing such patterns is key to determining the general rule that defines the sequence. Once you observe a consistent pattern, you can express it in a formula that captures the entire sequence in a single expression.
Mathematical Expressions
Turning patterns into mathematical expressions allows us to generalize and predict future values in a sequence. A mathematical expression is a combination of numbers, variables, operators, and functions that stands for a particular quantity.
In our sequence, 4, 8, 16, 32, and 64, each number can be written as a power of 2:
  • 4 is written as \(2^2\)
  • 8 is written as \(2^3\)
  • 16 is written as \(2^4\)
  • 32 is written as \(2^5\)
  • 64 is written as \(2^6\)
This repetition of powers suggests a formula for the sequence is\(s_{n-1} = 2^{n+1}\), which means that each term is the power of 2 raised to the variable \(n+1\).
This formula can be used to find any term in the sequence by substituting \(n\) with the term's position minus one.
Powers of Numbers
Understanding powers (or exponents) is crucial in sequences where terms are calculated using repeated multiplication.
Powers of numbers are used to denote multiplication of a number by itself a certain number of times. For example, \(2^3 = 2 \times 2 \times 2 = 8\).
In the given sequence:
  • The expression \(2^2\) equals 4
  • \(2^3\) equals 8
  • \(2^4\) equals 16
And it continues in this manner.
The sequence can be seen as powers of 2, growing exponentially as the sequence progresses, illustrating how exponents allow us to efficiently express large numbers and patterns in sequences.
This exponential growth factor is why it's so efficient to express the sequence using powers, particularly in geometric sequences.

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Most popular questions from this chapter

True or false. Give an explanation for your answer. \(\sum C_{n}(x-1)^{n}\) and \(\sum C_{n} x^{n}\) have the same radius of convergence.

Decide if the statements are true or false. Give an explanation for your answer. If \(b_{n} \leq a_{n} \leq 0\) for all \(n\) and \(\sum b_{n}\) converges, then \(\sum a_{n}\) converges.

Decide if the statements are true or false. Give an explanation for your answer. $$\text { If } \sum a_{n} \text { converges, then } \lim _{n \rightarrow \infty}\left|a_{n+1}\right| /\left|a_{n}\right| \neq 1$$

Decide if the statements are true or false. Give an explanation for your answer. If an alternating series converges, then the error in using the first \(n\) terms of the series to approximate the entire series is less in magnitude than the first term omitted.

For what values of \(a\) does the series converge? $$\sum_{n=1}^{\infty}(\ln a)^{n}, a>0$$
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