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Chapter 10: Problem 22

Write the formula for the \(n\) th term of each geometric series. $$a_{1}=1000 \quad r=0.5$$

Short Answer

Expert verified
The formula for the \( n \)-th term is \( a_n = 1000 \times 0.5^{n-1} \).

Step by step solution

01

Identify Known Components

The problem provides the first term of the geometric series, \( a_1 = 1000 \), and the common ratio, \( r = 0.5 \). These values will be used to write the general formula for the \( n \)-th term of a geometric series.
02

Recall Formula for the n-th Term

The \( n \)-th term of a geometric series can be calculated with the formula \( a_n = a_1 imes r^{n-1} \). This formula takes the first term and multiplies it by the common ratio raised to the power of \( n-1 \).
03

Substitute Known Values into the Formula

Substitute the known values \( a_1 = 1000 \) and \( r = 0.5 \) into the general formula: \( a_n = 1000 imes 0.5^{n-1} \).
04

Simplify the Expression

The expression \( 1000 imes 0.5^{n-1} \) is already in its simplest form for a formula, representing the \( n \)-th term of the given geometric series.

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Key Concepts

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

n-th term formula
The formula for the \( n \)-th term of a geometric sequence is a crucial tool. It allows us to quickly find any term in the sequence without having to calculate all the preceding terms. This is particularly helpful when dealing with long or infinite series. The formula is given by:\[ a_n = a_1 \, \cdot \, r^{n-1} \]In this equation:
  • \( a_n \) represents the \( n \)-th term you wish to find.
  • \( a_1 \) is the first term of the sequence.
  • \( r \) is the common ratio.
  • \( n \) is the position of the term in the sequence.
The formula's beauty lies in its simplicity. You start with the first term \( a_1 \) and multiply it by the common ratio raised to the power of \( n-1 \). This exponent, \( n-1 \), accounts for the number of times you apply the common ratio to move from the first term to the \( n \)-th term.
common ratio
In a geometric sequence, the common ratio \( r \) is a constant factor that each term is multiplied by to get the next term. It plays a central role in forming the sequence:
  • If \( r > 1 \), the sequence grows exponentially with each term.
  • If \( 0 < r < 1 \), the sequence will progressively get smaller, converging towards zero.
  • If \( r = 1 \), each term in the sequence is the same as the first term.
  • If \( r < 0 \), the sequence will alternate in sign.
The common ratio is the essence of what makes a sequence geometric. Without it functioning consistently across all the terms, you wouldn't have regular spacing or multiplication, which is the defining trait of geometric sequences.
geometric sequence
A geometric sequence 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. It can be represented as:
  • The sequence starts with \( a_1 \).
  • The second term is \( a_1 \times r \).
  • The third term is \( a_1 \times r^2 \).
  • And so forth, with the \( n \)-th term being \( a_1 \times r^{n-1} \).
Understanding the nature of the geometric sequence is crucial, as it applies to many real-world situations, such as population growth, radioactive decay, and financial mathematics. The series can grow rapidly if the ratio is greater than 1, or shrink towards zero if the ratio is between 0 and 1. Recognizing the structure of a geometric sequence allows for effective and efficient problem solving.

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Most popular questions from this chapter

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