Problem 33 - Algebra and Trigonometry

Question

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$ \sum_{i=1}^{10} 5 \cdot 2^{i} $$

Step-by-Step Solution

Answer

The sum of that geometric series \( \sum_{i=1}^{10} 5 \cdot 2^{i} \) is 10240.

Step 1: Identify the parameters

The first term a is the coefficient of the first term in the sequence, here 5 * 2^1 = 10. The common ratio r is the base of the exponent, which here is 2. The number of terms n is the difference between the limits of the sum, which here is 10-1+1 = 10.

Step 2: Substitute the parameters into the formula

Now substitute these values into the formula: \( S_n = \frac{a(1-r^n)}{1-r} = \frac{10(1-2^{10})}{1-2} \)

Step 3: Simplify the formula

After simplifying the above expression, the result is obtained as \( S_{10} = \frac{10(1-1024)}{-1} = 10240 \)

Key Concepts

Geometric Sequence

A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

For example, in the sequence 2, 4, 8, 16, ..., the common ratio is 2. This means each term is twice as large as the previous one. This multiplication factor remains consistent throughout the sequence. Recognizing this pattern is crucial because it sets the foundation for calculating the sum of the sequence.

Series and Summation

A series is the sum of the terms of a sequence, which can be finite or infinite. Summation, indicated by the sigma symbol \( \Sigma \), is the process of adding a sequence of numbers. In the context of a geometric sequence, we often look for the sum of the first 'n' terms, which is known as a finite geometric series.

To find the sum of a geometric series, you can use the formula: \[ S_n = \frac{a(1-r^n)}{1-r} \] where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms to be added. The formula is derived from the concept that the sum of a series is equal to the first term plus the sum of the remaining terms.

Exponential Functions

Exponential functions are closely related to geometric sequences. An exponential function is a mathematical function of the form \( f(x) = a \cdot b^x \) where 'b' is a positive real number, and 'a' is a constant.

The link between geometric sequences and exponential functions lies in their growth rates. Both increase (or decrease) by a common factor: in a geometric sequence, each term grows by a common ratio, while in an exponential function, the value grows by a constant factor as 'x' increases. Understanding the behavior of exponential growth helps to comprehend how rapidly the terms of a geometric sequence can grow over time, which is vital when working with series and their sums.