Problem 15

Question

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio. $$ a_{1}=5, r=\frac{1}{5} $$

Step-by-Step Solution

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Answer

The first five terms are 5, 1, $ \frac{1}{5} $, $ \frac{1}{25} $, and $ \frac{1}{125} $.

1Step 1: Determine the Formula for the nth Term

The nth term of a geometric sequence can be determined using the formula: $ a_n = a_1 imes r^{(n-1)} $ where $ a_1 $ is the first term and $ r $ is the common ratio.

2Step 2: Calculate the First Term

Substitute $ n = 1 $ into the formula to find the first term: $ a_1 = 5 $.

3Step 3: Calculate the Second Term

Substitute $ n = 2 $ into the formula: $ a_2 = 5 \times \left(\frac{1}{5}\right)^{1} = 1 $.

4Step 4: Calculate the Third Term

Substitute $ n = 3 $ into the formula: $ a_3 = 5 \times \left(\frac{1}{5}\right)^{2} = \frac{1}{5} $.

5Step 5: Calculate the Fourth Term

Substitute $ n = 4 $ into the formula: $ a_4 = 5 \times \left(\frac{1}{5}\right)^{3} = \frac{1}{25} $.

6Step 6: Calculate the Fifth Term

Substitute $ n = 5 $ into the formula: $ a_5 = 5 \times \left(\frac{1}{5}\right)^{4} = \frac{1}{125} $.

Key Concepts

nth term formulacommon ratiofirst term calculationsequence terms calculation

nth term formula

In the realm of geometric sequences, understanding the nth term formula is crucial. This formula helps you determine any term in the sequence without needing to calculate all previous terms. The nth term in a geometric sequence is given by the formula:\[a_n = a_1 \times r^{(n-1)}\]where:

$a_n$ is the nth term of the sequence,
$a_1$ is the first term of the sequence,
$r$ is the common ratio,
and $n$ is the term number you wish to find.

Using the formula allows you to directly jump to the term you are interested in, which is beneficial for solving problems quickly and accurately. You simply need to know the first term and the common ratio to unlock the entire sequence. By substituting different values for $n$, you can generate specific terms in the sequence easily.

common ratio

The common ratio is a fundamental characteristic of a geometric sequence. It is the factor by which we multiply to get from one term to the next. Understanding the common ratio is key to unraveling the pattern of the sequence. To find the common ratio $r$, examine how terms progress:\[r = \frac{a_n}{a_{n-1}}\]This equation lets you divide any term by the term preceding it to find $r$. For example, if you divide the second term by the first term or the third term by the second term, you should always get the same common ratio. Given a sequence where $a_1 = 5$ and $r = \frac{1}{5}$, each subsequent term is shaped by multiplying the previous term by $\frac{1}{5}$. Using a consistent pattern provides a reliable method to predict and extend sequence terms endlessly.

first term calculation

In any geometric sequence, the first term $a_1$ is the initial building block from which all other terms are derived. Knowing the first term is essential for applying the sequence formula accurately.The first term is typically provided in a problem or can be identified directly from the sequence. For instance, in this exercise, we're given $a_1 = 5$. This specific term serves as the starting point for computing all successive terms using the nth term formula. Simply put, you:

Identify or receive the first term,
Utilize this term in conjunction with the common ratio to find any other term in the sequence.

Starting with the correct first term ensures that the outputs for each sequential term calculated from the formula align precisely with the geometric progression's structure.

sequence terms calculation

Calculating the terms of a geometric sequence is straightforward once you understand the relationship between the terms, the first term, and the common ratio. Let's detail a few steps by using the initial example:

Start with the first term: $a_1 = 5$.
Use the nth term formula to find each subsequent term.

Second term:\[a_2 = 5 \times \left(\frac{1}{5}\right)^{1} = 1\]Third term:\[a_3 = 5 \times \left(\frac{1}{5}\right)^{2} = \frac{1}{5}\]Continue similarly for additional terms:Fourth term:\[a_4 = 5 \times \left(\frac{1}{5}\right)^{3} = \frac{1}{25}\]Fifth term:\[a_5 = 5 \times \left(\frac{1}{5}\right)^{4} = \frac{1}{125}\]Each step involves raising the common ratio to the power of $n-1$ and multiplying by the first term. By following these steps, you can quickly generate the terms of any geometric sequence.

Problem 15

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