Problem 4 - Beginning and Intermediate Algebra

Question

Fill in each blank with the correct response. The number of terms in the geometric sequence \(1,2,4, \ldots, 2048\) is ______.

Step-by-Step Solution

Answer

There are 12 terms in the sequence.

Step 1 - Identify the first term and common ratio

In a geometric sequence, the first term is denoted by \(a\). For the given sequence, the first term \(a = 1\). The common ratio \(r\) is the ratio between the successive terms, which can be found by dividing the second term by the first term: \(r = \frac{2}{1} = 2\).

Step 2 - Recall the formula for the nth term of a geometric sequence

The nth term of a geometric sequence can be found using the formula: \(a_n = a \cdot r^{(n-1)}\). In this formula, \(a_n\) is the nth term, \(a\) is the first term, and \(r\) is the common ratio.

Step 3 - Set up the equation with the given nth term

Substitute the values into the formula where the nth term \(a_n = 2048\), the first term \(a = 1\), and the common ratio \(r = 2\). The equation becomes: \[2048 = 1 \cdot 2^{(n-1)}\].

Step 4 - Solve for \(n\)

To find the number of terms \(n\), solve the equation \(2048 = 2^{(n-1)}\). Take the logarithm of both sides or recognize that 2048 is a power of 2. Knowing that \(2048 = 2^{11}\), we set the equation: \[2^{11} = 2^{(n-1)}\]. Equating the exponents, we get: \[11 = n-1\]. Solving for \(n\), we get: \[n = 11 + 1 = 12\].

Key Concepts

first term

In a geometric sequence, the first term is the starting point of the sequence. It's typically denoted by the letter \(a\).
This term is crucial because it's the baseline from which all other terms in the sequence are calculated.
For example, in the sequence given in our exercise \(1, 2, 4, \ldots, 2048\), the first term \(a\) is 1. This value sets the stage for all subsequent terms by providing a reference point.

common ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term.
It's denoted by the letter \(r\). To find the common ratio, you can divide any term in the sequence by the term that directly precedes it.
In our case, for the sequence \(1, 2, 4, \ldots, 2048\), the common ratio can be found by dividing the second term by the first term: \( \frac{2}{1} = 2 \). Thus, our common ratio \(r\) is 2.

• This ratio is consistent throughout the sequence.
• It's crucial for determining any term in the sequence based on its position.

nth term formula

The nth term formula for a geometric sequence helps us find any term in the sequence without listing all previous terms.
The formula is given by \(a_n = a \cdot r^{(n-1)}\), where:

• \(a_n\) is the nth term.
• \(a\) is the first term.
• \(r\) is the common ratio.

Using this formula simplifies calculations, especially for large sequences.
For example, if you want to find the 5th term in our sequence:

• The first term \(a\) is 1.
• The common ratio \(r\) is 2.
• So the 5th term would be \(1 \cdot 2^{(5-1)} = 1 \cdot 2^4 = 16\).

Understanding and applying the nth term formula is key to solving problems related to geometric sequences efficiently.