Problem 46 - Precalculus Mathematics for Calculus

Question

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$a_{2}=0.12, \quad a_{5}=0.00096, \quad n=4$$

Step-by-Step Solution

Answer

The partial sum \(S_4\) is approximately 0.7488.

Step 1: Determine the common ratio

To find the common ratio \(r\), we use the formula for the \(k\)-th term of a geometric sequence: \(a_k = a_1 \cdot r^{k-1}\). We know \(a_2 = 0.12\) and \(a_5 = 0.00096\). We express \(a_2\) and \(a_5\) in terms of \(a_1\): \[ a_2 = a_1 \cdot r = 0.12 \] \[ a_5 = a_1 \cdot r^4 = 0.00096 \]. Divide the equation for \(a_5\) by \(a_2\) to eliminate \(a_1\): \[ \frac{a_5}{a_2} = \frac{a_1 \cdot r^4}{a_1 \cdot r} = r^3. \] Thus, \(r^3 = \frac{0.00096}{0.12}\), and \(r^3 = 0.008\). Solve for \(r\): \[ r = \sqrt[3]{0.008} = 0.2. \] So, the common ratio \(r\) is \(0.2\).

Step 2: Determine the first term

With \(r = 0.2\), use \(a_2 = a_1 \cdot r = 0.12\) to find \(a_1\): \[ a_1 \cdot 0.2 = 0.12 \rightarrow a_1 = \frac{0.12}{0.2} = 0.6. \] So, the first term \(a_1\) is \(0.6\).

Step 3: Use the formula for the partial sum of the geometric sequence

The formula for the partial sum \(S_n\) of the first \(n\) terms of a geometric sequence is: \[ S_n = a_1 \frac{1 - r^n}{1 - r}. \] Substitute \(a_1 = 0.6\), \(r = 0.2\), and \(n = 4\) into this formula: \[ S_4 = 0.6 \cdot \frac{1 - (0.2)^4}{1 - 0.2}. \] Simplify the expression: \[ S_4 = 0.6 \cdot \frac{1 - 0.0016}{0.8} = 0.6 \cdot \frac{0.9984}{0.8}. \] Calculate \(S_4\): \[ S_4 = 0.6 \cdot 1.248 = 0.7488. \] Therefore, the partial sum \(S_4\) is approximately \(0.7488\).

Key Concepts

Partial Sum

A partial sum is the sum of a specified number of terms in a sequence. In a geometric sequence, this is particularly valuable as it allows you to find the total of the first few terms rapidly. The formula for the partial sum, \( S_n \), of the first \( n \) terms of a geometric sequence is:

• \( S_n = a_1 \frac{1 - r^n}{1 - r} \) for \( r eq 1 \)

Here, \( a_1 \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the number of terms you are adding together.
It's crucial to note that this formula only works when \( r eq 1 \), since a common ratio of 1 would involve division by zero, which is undefined.
Using this formula, you can effectively sum up the terms without individually adding each one, saving time and reducing error.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next one. It is a fundamental component to identify in any geometric sequence.
To find the common ratio, divide any term in the sequence by the previous term, provided it is not the first.
In our example, we had two terms: \( a_2 = 0.12 \) and \( a_5 = 0.00096 \). By expressing both terms in relation to the first term and equating the two, we can derive:
\[ r^3 = \frac{0.00096}{0.12} = 0.008 \]
Then solve for \( r \) by taking the cube root:\
\[ r = \sqrt[3]{0.008} = 0.2 \]
This common ratio tells us that each term is 20% of the previous one, resulting in terms decreasing in size rapidly.
Understanding the common ratio helps in predicting any term in the sequence without direct computation of all preceding terms.

First Term

The first term of a geometric sequence, denoted as \( a_1 \), sets the stage for all subsequent terms. It is the initial value from which the sequence builds or decays, based on the common ratio.
In our exercise, after finding the common ratio \( r = 0.2 \), determining the first term \( a_1 \) becomes straightforward by using a known term:
Starting with \( a_2 = a_1 \cdot r = 0.12 \), you can rearrange to find \( a_1 \):
\[ a_1 = \frac{0.12}{0.2} = 0.6 \]
The first term is essentially the cornerstone of the sequence - the basis from which all other terms are derived through multiplication by the common ratio.
Knowing \( a_1 \) is vital for calculating any specific term or the partial sum in the sequence.