Question

In Exercises 47–52, find the sum. $$ \sum_{i=1}^{10} 7(4)^{i-1} $$

Short answer

The short answer is -7669583421 after evaluating the expression of Step 3.

Step-by-step solution

Identify a, r, n

Identify 'a' (the first term in the sequence), 'r' (the ratio of the sequence), and 'n' (the number of terms in the series). In this case, 'a' is 7, as that is the first term in the series. 'r' is 4, as that's the number each term is multiplied by to get the next term in the sequence. 'n' is 10 since there are ten terms in the series.

Calculate the sum

To calculate the sum of the series, use the formula for the sum of a geometric series: \[ \sum = a * (\frac{1-r^n}{1-r}) \] Inserting the values, we get \[ \sum = 7 * (\frac{1-4^{10}}{1-4}) \]

Evaluate the expression

After calculating the above expression, we'll find the value of the sum.

Key concepts

Geometric Progression

Understanding a geometric progression is central when dealing with series and sequences that rise or fall exponentially. A geometric progression is a sequence of numbers where each successive term is obtained by multiplying the previous one by a fixed, non-zero number known as the common ratio. This characteristic ratio is what gives the progression its exponential nature.

Let's consider the example \( 7, 28, 112, 448, ... \), where the common ratio \( r \) is 4, because each term is 4 times the preceding term. This relationship is exponential because each term grows at a rate proportional to its current value, creating a swift increase or decrease in figure depending on whether the ratio is greater than or less than one.

Series and Sequences

When we take elements from a sequence and sum them together, this creates a series. A sequence is simply an ordered list of numbers following a specific pattern. As such, a geometric series is the sum of the terms of a geometric progression. Consider the sequence \( 7, 28, 112, ... \); its corresponding series would be \( 7 + 28 + 112 + ... \), and so forth.

In mathematics, understanding series and sequences is essential as they frequently model real-world scenarios. For instance, compound interest in finance or radioactive decay in physics can be modeled using geometric series. Learning to calculate the sum of such series not only serves academic purposes but also applications in various professional fields.

Exponential Expressions

Exponential expressions, as encountered in our example series \( \sum_{i=1}^{10} 7(4)^{i-1} \), are powerful mathematical tools to describe growth or decay. The expression \( 4^{i-1} \) represents an exponential function, where the base 4 is raised to the power of \( (i-1) \), with \( i \) being the position term of the series. What's particularly intriguing about exponential expressions is their compounded effect, which can yield very large or very small numbers based on the exponent value. This concept is widely used in various scientific calculations, like population growth or the spread of a virus, where the rate of increase is proportional to the current amount at each successive point in time.

Using exponential expressions in the context of geometric series sums allows us to evaluate and understand how quantities are compounded over multiple periods, giving us insight into predictable patterns of exponential growth or decay.