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Chapter 9: Problem 14

Write the first five terms of the geometric sequence. $$a_{1}=8, r=2$$

Short Answer

Expert verified
The first five terms of the given geometric sequence are: 8, 16, 32, 64, 128

Step by step solution

01

Calculate the second term

Use formula for the nth term of a geometric sequence. Substituting \(a_{1} = 8\), \(r = 2\), and \(n = 2\) into the formula: \(a_{2} = 8 \cdot 2^{(2 - 1)} = 16\
02

Calculate the third term

Again use the formula and replace \(a_{1} = 8\), \(r = 2\), and \(n = 3\), you get: \(a_{3} = 8 \cdot 2^{(3 - 1)} = 32\
03

Calculate the fourth term

Substitute the values \(a_{1} = 8\), \(r = 2\), and \(n = 4\) into the formula: \(a_{4} = 8 \cdot 2^{(4 - 1)} = 64\
04

Calculate the fifth term

Use the same formula and replace \(a_{1} = 8\), \(r = 2\), and \(n = 5\), you obtain: \(a_{5} = 8 \cdot 2^{(5 - 1)} = 128\
05

Gathering all the terms

Collect all the terms to form the first five terms of the geometric sequence: 8, 16, 32, 64, 128

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Key Concepts

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

Sequence Terms
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For the given exercise, we are asked to find the first five terms of a geometric sequence starting with the first term, denoted as \(a_1\), which is 8.
In this sequence, each term builds upon the one before it by a process of multiplication. This means that if you understand how to find one term, you can find the next simply by using the common ratio to multiply.
  • The first term \(a_1\) is 8.
  • For the second term \(a_2\), multiply the first term by the common ratio.
  • Continue multiplying each subsequent term to determine \(a_3\), \(a_4\), and so on until you have the desired number of terms.
Thus, for the sequence in the exercise, the terms are calculated as follows: 8, 16, 32, 64, and 128.
Common Ratio
The common ratio is a key element in understanding geometric sequences. It is the constant factor between consecutive terms in the sequence. In this exercise, the common ratio is denoted by \(r\) and is given as 2.
To determine the subsequent terms within the sequence, multiply the preceding term by the common ratio. This multiplication rule helps maintain the geometric characteristic of the sequence.
  • For the sequence we鈥檙e working with, starting with a common ratio \(r = 2\), the next term is always double the previous term.
  • This makes it easy to predict future terms once you know a few initial terms and the ratio.
  • For example, beginning with the first term \(a_1 = 8\), the second term \(a_2\) becomes \(8 \times 2 = 16\).
Using the common ratio effectively allows you to determine the sequence without knowing each term individually at the start.
Nth Term
To find any term in a geometric sequence, using a specific formula is crucial. The nth term of a geometric sequence can be calculated using the formula: \(a_n = a_1 \cdot r^{(n-1)}\). Here, \(a_n\) represents the term you are solving for, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
For example, in the sequence where the first term \(a_1\) is 8 and the common ratio \(r\) is 2:
  • To find the second term \(a_2\), set \(n = 2\) and use the formula: \(a_2 = 8 \cdot 2^{(2-1)} = 16\).
  • To find the third term \(a_3\), set \(n = 3\): \(a_3 = 8 \cdot 2^{(3-1)} = 32\).
  • This formula can predict any term you need, such as \(a_4 = 8 \cdot 2^{(4-1)} = 64\), and so forth.
Utilizing the formula ensures that any term in the sequence can be calculated efficiently, whether it鈥檚 the fifth term as in the exercise or the 50th.

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