Problem 50 - Precalculus Mathematics for Calculus

Question

Find the sum. $$\sum_{j=0}^{5} 7\left(\frac{3}{2}\right)^{j}$$

Step-by-Step Solution

Answer

The sum is 145.46875.

Step 1: Understand the Expression

The given expression is a summation: \(\sum_{j=0}^{5} 7\left(\frac{3}{2}\right)^{j}\). This means we are summing the terms generated by the function \(7\left(\frac{3}{2}\right)^{j}\) as \(j\) runs from 0 to 5.

Step 2: Identify the Terms

The terms of the series are obtained by substituting values of \(j\) from 0 to 5: \(7\left(\frac{3}{2}\right)^{0}, 7\left(\frac{3}{2}\right)^{1}, 7\left(\frac{3}{2}\right)^{2}, \ldots, 7\left(\frac{3}{2}\right)^{5}\).

Step 3: Calculate Each Term

Compute each term:- For \(j = 0\), the term is \(7\left(\frac{3}{2}\right)^{0} = 7\).- For \(j = 1\), it is \(7\left(\frac{3}{2}\right)^{1} = 7 \times \frac{3}{2} = 10.5\).- For \(j = 2\), \(7\left(\frac{3}{2}\right)^{2} = 7 \times \frac{9}{4} = 15.75\).- For \(j = 3\), \(7\left(\frac{3}{2}\right)^{3} = 7 \times \frac{27}{8} = 23.625\).- For \(j = 4\), \(7\left(\frac{3}{2}\right)^{4} = 7 \times \frac{81}{16} = 35.4375\).- For \(j = 5\), \(7\left(\frac{3}{2}\right)^{5} = 7 \times \frac{243}{32} = 53.15625\).

Step 4: Sum the Terms

Add all the calculated terms: \(7 + 10.5 + 15.75 + 23.625 + 35.4375 + 53.15625\).

Step 5: Calculate the Final Sum

The sum is calculated as follows:\[ 7 + 10.5 + 15.75 + 23.625 + 35.4375 + 53.15625 = 145.46875 \]

Step 6: Conclusion

The final result of the sum \(\sum_{j=0}^{5} 7\left(\frac{3}{2}\right)^{j}\) is calculated as 145.46875.

Key Concepts

Summation

Summation is an essential concept in mathematics. It involves adding up a sequence of numbers. Understanding summation helps solve various problems, especially when dealing with series.
Summation is often represented using the symbol \(\Sigma\). This symbol tells us to sum a sequence based on a given rule. In our exercise, the summation \(\sum_{j=0}^{5} 7\left(\frac{3}{2}\right)^{j}\) means we need to add multiple terms.
This particular example starts from \(j = 0\) and goes up to \(j = 5\). Each term follows a pattern defined by the expression \(7\left(\frac{3}{2}\right)^{j}\). As we plug in different values of \(j\), we generate terms that we then add together.

• Recognizing the index of summation \(j\) is vital.
• The starting and ending values define how many terms we'll have.
• Each term is generated by replacing \(j\) in the given formula.

Mastering summation is key for understanding series, especially in calculus and other areas of advanced math.

Exponential Function

In mathematics, exponential functions describe processes where growth or decay is proportional to current value. They have the form \(a^x\), where \(a\) is the base, and \(x\) is the exponent. Here, we see this concept in the expression \(\left(\frac{3}{2}\right)^j\).
Exponential growth depicts how quickly something can rise. If you raise a number greater than 1 to increasing powers, it expands swiftly. This concept appears often in real-world scenarios, like population growth or compound interest.

• The exponential function is particularly useful because it grows at an accelerating rate.
• Understanding the base and exponent role is crucial. The base dictates the rate of growth or decay.
• In our exercise, \(\left(\frac{3}{2}\right)^j\) is the growing component of each term.

Handling exponential functions helps identify rapid changes, making it useful in calculating series, especially geometric ones as in this exercise.

Series Calculation

Calculating a series involves adding together terms of a sequence. In our example, we're dealing with a geometric series, where each term contains a constant ratio between consecutive terms. The formula \(7\left(\frac{3}{2}\right)^{j}\) suggests that it's geometric.
Here's how series calculation works in steps:
Each term is calculated separately. You start by plugging in the smallest value for \(j\), i.e., 0, and work your way up to the largest value, which is 5 in this scenario.

• The first term is simply 7, since \(\left(\frac{3}{2}\right)^0 = 1\).
• As \(j\) increases, the effect of raising \(\frac{3}{2}\) to \(j\) becomes larger, enhancing each subsequent term.
• Finally, sum all calculated values: 7, 10.5, 15.75, etc.

This approach gives the total sum of the series: 145.46875. Series calculations are vital in understanding accumulated values over sequences, notably in applications like computing interest or analyzing patterns.