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Chapter 11: Problem 22

Find the indicated term for each geometric sequence. \(2,8,32, \ldots ;\) the 9 th term

Short Answer

Expert verified
The 9th term is 131072.

Step by step solution

01

Identify the first term (a)

The first term of the sequence is given as 2. So, let鈥檚 denote it by \(a\).
02

Determine the common ratio (r)

Calculate the common ratio \(r\) by dividing the second term by the first term. \(r = \frac{8}{2} = 4\).
03

Use the nth term formula

The formula for the nth term in a geometric sequence is \(a_n = a \times r^{(n-1)}\).
04

Plug in the values

Substitute \(a = 2\), \(r = 4\), and \(n = 9\) into the formula: \(a_9 = 2 \times 4^{(9-1)} = 2 \times 4^8\).
05

Calculate the power

Compute \(4^8\): \(4^8 = 65536\).
06

Multiply

Finally, multiply the first term by the computed power: \(a_9 = 2 \times 65536 = 131072\).

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Key Concepts

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

First Term
In a geometric sequence, the first term is the initial number of the series.
The first term is crucial because it sets the stage for the sequence that follows. Think of it as the starting point.
In our example, the first term is 2. We denote it by the letter 'a'.
So, for the given sequence:
  • First term (\(a\)) = 2
Understanding the first term helps us in accurately calculating other terms in the sequence.
Common Ratio
The common ratio in a geometric sequence is the factor by which each term is multiplied to get the next term.
It determines the 'growth' or 'decay' rate of the sequence.
To find the common ratio, we divide the second term by the first term.
For our series, that looks like this:
  • Common ratio (\(r\)) = \( \frac{8}{2} = 4 \)
The ratio is constant throughout the sequence, providing the consistent multiplication factor.
nth Term Formula
To find any term in a geometric sequence, we use the nth term formula.
It allows us to pinpoint the exact value of any term without having to list out all the terms.
Here is the formula:
\( a_n = a \times r^{(n-1)} \)
  • \(a\) is the first term
  • \(r\) is the common ratio
  • \(n\) is the term number
This formula gives us a straightforward way to calculate any term in the sequence.
Power Calculation
When using the nth term formula, calculating the power is crucial.
It involves raising the common ratio to a certain exponent.
From our example, to find the 9th term, we substitute values into the formula:
  • \( a_9 = 2 \times 4^{(9-1)} = 2 \times 4^8 \)
Next, we calculate the power:
  • \(4^8 = 65536\)
Finally, multiply the first term (\(a\)) by this value:
  • \(2 \times 65536 = 131072\)
So, the 9th term in the sequence is 131072.
Power calculations can be tricky, but with practice, they become more manageable.

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