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Chapter 11: Problem 179

Write an explicit formula for each geometric sequence. \(a_{n}=\\{-2,-4,-8,-16, \ldots\\}\)

Short Answer

Expert verified
The explicit formula is \\(a_n = (-2) \cdot 2^{(n-1)}\\).

Step by step solution

01

Identify the Common Ratio

A geometric sequence is defined by a constant ratio between consecutive terms. Here, the sequence is \(-2, -4, -8, -16, \ldots\). To find the common ratio ( ), divide the second term by the first term: \(-4 梅 -2 = 2\). Similarly, \(-8 梅 -4 = 2\). Thus, the common ratio \(r = 2\).
02

Identify the First Term

The first term of the sequence is the value of \(a_1\). From the sequence \(-2, -4, -8, -16, \ldots\), the first term is \(-2\). Therefore, \(a_1 = -2\).
03

Write the Explicit Formula

The explicit formula for a geometric sequence is given by \((a_n = a_1 \cdot r^{(n-1)})\) where \(a_1\) is the first term and \(r\) is the common ratio. Substituting the known values gives \(a_n = (-2) \cdot 2^{(n-1)}\).

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Key Concepts

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

Explicit Formula
In the world of geometric sequences, the explicit formula is a powerful tool. It allows you to find any term at any position without having to list out all previous terms. For a geometric sequence, this formula is written as \[ a_n = a_1 \cdot r^{(n-1)} \] where:
  • \( a_n \) is the term you wish to find
  • \( a_1 \) is the first term
  • \( r \) is the common ratio
  • \( n \) is the position of the term in the sequence

This formula is particularly handy because it condenses the whole sequence into a single equation. So instead of noting down the entire sequence to solve for a certain term, you can simply plug in the values you know and find the answer directly. This simplification not only saves time but also minimizes errors.
Common Ratio
The common ratio in a geometric sequence is what makes it distinctively 'geometric'. It is the factor by which we multiply one term to get to the next. In the given sequence \(-2, -4, -8, -16, \ldots\), you find the common ratio \( r \) by dividing a term by its immediate predecessor. For instance:
  • \(-4 \div -2 = 2\)
  • \(-8 \div -4 = 2\)

The common ratio here is consistently \( r = 2 \). If you continue dividing any subsequent pairs of terms, the ratio will remain 2. This constant multiplier (the common ratio) ensures the sequence is geometric. It is crucial in forming the explicit formula and predicting future terms.
First Term
The first term of a geometric sequence is the starting point of the entire series. It is denoted as \( a_1 \). This initial term is fundamental for determining any other term in the sequence through the explicit formula.Consider the sequence \(-2, -4, -8, -16, \ldots\). The very first term is \(-2\), hence \( a_1 = -2 \).
Thinking about the first term is like setting the starting block in a race. It establishes the base value for the entire sequence to follow. Without identifying \( a_1 \), plugging numbers into the explicit formula would be ineffective, as we would lack a starting reference.

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