Problem 50 - Algebra 2

Question

Find the sum of each geometric series. $$ \sum_{n=1}^{16} 4 \cdot 3^{n-1} $$

Step-by-Step Solution

Answer

The sum is 86,093,440.

Step 1: Identify the Series

The given series can be expressed as \( a_n = 4 imes 3^{n-1} \) where \( n \) ranges from 1 to 16. We recognize this as a geometric series because each term is obtained by multiplying the previous term by a common ratio. Here, the first term \( a_1 = 4 \) and the common ratio \( r = 3 \).

Step 2: Formula for Sum of a Geometric Series

The sum \( S \) of the first \( n \) terms of a geometric series with first term \( a \), common ratio \( r \), and total terms \( n \) is given by \[ S_n = a \frac{r^n - 1}{r - 1}. \] We'll use this formula to find the sum of the series from \( n = 1 \) to \( n = 16 \).

Step 3: Substitute Known Values into the Formula

Substitute \( a = 4 \), \( r = 3 \), and \( n = 16 \) into the formula. The equation becomes \[ S_{16} = 4 \frac{3^{16} - 1}{3 - 1} = 4 \frac{3^{16} - 1}{2}. \]

Step 4: Calculate \( 3^{16} \)

Calculate \( 3^{16} \). Using a calculator, \( 3^{16} = 43,046,721 \). Substitute this value back into the equation.\[ S_{16} = 4 \frac{43,046,721 - 1}{2}. \]

Step 5: Simplify the Expression

First calculate \( 43,046,721 - 1 = 43,046,720 \). Then, simplify the expression:\[ S_{16} = 4 \frac{43,046,720}{2} = 4 imes 21,523,360. \]

Step 6: Calculate the Final Result

Multiply \( 4 imes 21,523,360 \) to find the sum of the series. \[ S_{16} = 86,093,440. \]

Step 7: Conclusion

The sum of the geometric series \( \sum_{n=1}^{16} 4 \cdot 3^{n-1} \) is obtained as \( 86,093,440 \).

Key Concepts

Sum of a geometric series

A geometric series consists of the sum of terms in a geometric sequence. This means each term in the series is a multiple of the previous term, with this multiplier known as the common ratio. When faced with a geometric series problem, like the one given, we can use a handy formula to calculate the series' sum. This formula is:

• \( S_n = a \frac{r^n - 1}{r - 1} \)

Here, \( S_n \) represents the sum of the first \( n \) terms. \( a \) is the first term of the series, \( r \) is the common ratio, and \( n \) is the number of terms. Using this formula not only simplifies finding the sum but also makes calculations more straightforward than adding each term individually.
In our example, the first term is 4, and the common ratio is 3, with the series stretching out to 16 terms. By inputting these values into the formula, we've calculated that the sum of this series is \( 86,093,440 \), showcasing the power and simplicity of using the formula for geometric series.

Common ratio

The common ratio is a fundamental component of any geometric sequence or series. It is the constant factor by which we multiply each term to get the next term in the sequence. In simple terms, it is what 'connects' each step in the series or sequence. In our exercise, the common ratio is 3.
To find the common ratio in a sequence given its terms, we divide any term by the previous one. Consistency in this division across the sequence confirms it as the common ratio.

• If \( a_1 = 4 \) (the first term) and \( a_2 = 12 \) (the second term), the common ratio \( r \) would be calculated as \( \frac{12}{4} = 3 \).
• Knowing the common ratio allows us to predict future terms and also use the sum formula effectively.

Geometric sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. The sequence's initial term is denoted by \( a_1 \) and the common ratio by \( r \).
The formula for the nth term of a geometric sequence can be expressed as:

• \( a_n = a_1 \cdot r^{(n-1)} \)

This formula helps us find any specific term in the sequence without listing all preceding terms. For instance, in our example with a first term \( a_1 = 4 \) and a common ratio \( r = 3 \), the 16th term can be calculated using:

• \( a_{16} = 4 \cdot 3^{(16-1)} \)

Geometric sequences are valuable in diverse fields, from finance to physics, for modeling growth or decay processes. Understanding the geometric sequence's behavior from its initial terms and common ratio can unlock insights into such patterns.