Problem 52 - 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 0.7488.

Step 1: Identify the terms of the sequence

The problem provides the second term \(a_2 = 0.12\) and the fifth term \(a_5 = 0.00096\) of a geometric sequence. Use these terms to find the first term \(a\) and the common ratio \(r\).

Step 2: Use the formula for the n-th term in a geometric sequence

For any term in a geometric sequence, we can use the formula \(a_n = ar^{n-1}\). For \(a_2\), we have \(0.12 = ar^{2-1} = ar\), and for \(a_5\), we have \(0.00096 = ar^{5-1} = ar^4\).

Step 3: Solve for common ratio \(r\)

Divide the equation for \(a_5\) by the equation for \(a_2\): \(\frac{0.00096}{0.12} = \frac{ar^4}{ar} = r^3\). Simplify this to get \(r^3 = \frac{0.00096}{0.12} = 0.008\). Thus, \(r = 0.2\).

Step 4: Find first term \(a\)

With \(r = 0.2\), substitute back into one of the original equations, \(0.12 = ar\). This gives \(ar = 0.12\) so the first term \(a = \frac{0.12}{0.2} = 0.6\).

Step 5: Calculate the partial sum \(S_n\) for \(n=4\)

Use the formula for the partial sum of a geometric sequence: \(S_n = a \frac{1-r^n}{1-r}\). Substitute \(a=0.6\), \(r=0.2\), and \(n=4\) to get:\[S_4 = 0.6 \frac{1-(0.2)^4}{1-0.2} = 0.6 \frac{1-0.0016}{0.8}\].

Step 6: Simplify the expression for \(S_n\)

Calculate \(1 - (0.2)^4 = 1 - 0.0016 = 0.9984\) and then simplify to find:\[S_4 = 0.6 \frac{0.9984}{0.8} = 0.6 \times 1.248 = 0.7488\].

Key Concepts

Partial Sums

In any geometric sequence, finding the partial sum means calculating the sum of a certain number of terms in that sequence. It's like adding part of the series rather than the whole thing. The formula to find the partial sum of a geometric sequence is helpful when needing the cumulative total of the first few terms, say the first 'n' terms.
For a geometric sequence, the partial sum formula is given by:

• \( S_n = a \frac{1-r^n}{1-r} \)

where:

• \( S_n \) is the partial sum of the first \( n \) terms
• \( a \) is the first term of the sequence
• \( r \) is the common ratio
• \( n \) is the number of terms to be summed

Understanding this formula involves simply plugging in the known values for each variable—like we did earlier, by setting \( a = 0.6 \), \( r = 0.2 \), and \( n = 4 \), which yields the result \( S_4 = 0.7488 \). Easy peasy, right?

Common Ratio

A geometric sequence is defined by its common ratio. This is what makes each term relate to the next. In essence, each term is a multiple of the previous one by a constant factor, the common ratio.
To find the common ratio \( r \), use the formula connected to any two terms of a geometric sequence:

• \( r = \frac{a_{n+k}}{a_n} \)

This formula tells us the ratio between any two consecutive terms. But in some cases, like in the exercise, we may know non-consecutive terms. So, we need our formulas to reflect the gap between them, which is why we use \( r^k \) where \( k \) is the difference in term positions. Given \( a_2 = 0.12 \) and \( a_5 = 0.00096 \), by solving \( \frac{0.00096}{0.12} = r^3 \), we find that \( r = 0.2 \), meaning each term is just 0.2 of its preceding term in the sequence.
This factor's consistency is what defines the sequence's pace.

N-th Term Formula

One of the useful aspects of understanding sequences is being able to predict any term's value without listing all previous ones. This prediction uses what we call the "n-th term formula."
In a geometric sequence, any term \( a_n \) is given by:

• \( a_n = ar^{n-1} \)

This means every term depends on:

• the first term \( a \)
• the position of the term \( n \)
• the common ratio \( r \)

In our earlier problem, to find specific terms like \( a_2 \) or \( a_5 \), we rearranged this formula. First, knowing \( a_2 = ar = 0.12 \) helped us relate it to \( ar^4 = 0.00096 \) and find \( r = 0.2 \). Thus, this formula allows us to map the exponential growth or decay within the sequence depending upon whether \( r \) is greater than or less than 1.