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Chapter 10: Problem 24

Write the formula for the \(n\) th term of each geometric series. $$a_{1}=\frac{1}{200} \quad r=5$$

Short Answer

Expert verified
The nth term is \( a_n = \frac{1}{200} \times 5^{(n-1)} \).

Step by step solution

01

Identify the General Formula for Geometric Sequences

The general formula for the nth term of a geometric series is given by: \[ a_n = a_1 imes r^{(n-1)} \] where \( a_1 \) is the first term and \( r \) is the common ratio.
02

Substitute the Given Values

Substitute the given first term \( a_1 = \frac{1}{200} \) and the common ratio \( r = 5 \) into the general formula. Thus, the formula becomes: \[ a_n = \frac{1}{200} \times 5^{(n-1)} \]
03

Write the Final Formula

The nth term of the given geometric sequence is thus determined as: \[ a_n = \frac{1}{200} imes 5^{(n-1)} \] This formula gives the nth term of the geometric sequence based on the given first term and the common ratio.

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Key Concepts

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

Understanding the nth Term Formula in Geometric Sequences
The nth term formula is a crucial concept for anyone learning about geometric sequences. It helps us find any term in the sequence without needing to list all the terms up to that point.
The formula used to calculate the nth term, often denoted as \(a_n\), is:
  • \(a_n = a_1 \times r^{(n-1)}\)
In this formula:
  • \(a_n\) is the term we want to find.
  • \(a_1\) is the first term in the sequence.
  • \(r\) is the common ratio, which we will explore further in the next section.
  • \(n\) is the term number we're interested in.
With this formula, we can efficiently calculate any term in a geometric sequence once we have the first term and the common ratio. This significantly reduces the effort required for calculations.
Deciphering the Common Ratio
The common ratio is an essential element of any geometric sequence. It determines how each term in the sequence relates to the previous term.
Simply put, the common ratio, denoted by \(r\), is the factor by which you multiply one term in a sequence to get the next term. For example, if each term is five times the previous one, the common ratio \(r\) is 5.
  • In our formula and example, the common ratio is provided: \(r = 5\).
  • To find the common ratio in any sequence where it's not provided, divide any term by the term immediately preceding it.
The common ratio is responsible for the exponential nature of the sequence.
It's a constant multiplier, and regardless of which terms you pick to evaluate, it remains the same throughout the sequence.
Formulating the Sequence Formula
Formulating the sequence formula for any geometric sequence is an essential process. With both the first term and the common ratio determined, you can construct a specific sequence formula applicable to your series.
The sequence formula integrates the nth term formula and the known values of \(a_1\) and \(r\) to give a single expression:
  • \(a_n = \frac{1}{200} \times 5^{(n-1)}\)
By plugging in this formula, you can determine any desired term in the sequence quickly and accurately.
  • The sequence formula allows for flexibility as \(n\) can be any integer.
  • This formula captures the entire sequence's behavior through a single expression.
This efficient method keeps calculations manageable even if \(n\) is a large number, providing significant insights into the sequence's growth and characteristics.

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Most popular questions from this chapter

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