Problem 45 - Algebra and Trigonometry

Question

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 2+2^{2}+2^{3}+\cdots+2^{11} $$

Step-by-Step Solution

Answer

The sum \(2 + 2^{2} + 2^{3} + \cdots + 2^{11}\) can be expressed using summation notation as \(\sum_{i=1}^{11} 2^{i}\).

Step 1: Identify 'a', 'r', and 'n'

In the given series, the first term 'a' is \(2^{1} = 2\), the common ratio 'r' is also 2 (since we multiply the previous number by 2 each time), and the number of terms 'n' is 11 (as given in the problem as the highest power of 2).

Step 2: Write in summation notation

We use the standard formula of a geometric series, which can be rewritten in summation notation as \(\sum_{i=1}^{n} ar^{i-1}\). We substitute the identified 'a', 'r', and 'n': \(\sum_{i=1}^{11} 2 \cdot 2^{i-1}\). Since multiplying by 2 is the same as raising 2 to a power of 1. we can simplify the expression to \(\sum_{i=1}^{11} 2^{i}\).

Key Concepts

Geometric Series

Understanding the concept of a geometric series is essential for solving problems involving sequences that progress multiplicatively. A geometric series is a series of numbers each term of which is a constant multiple, known as the common ratio, of the previous term. This is crucial when dealing with interests in finance, physics involving exponentially decaying systems, and other real-life applications.

For instance, in the exercise \(2+2^{2}+2^{3}+\cdots+2^{11}\), we recognize a pattern where each term after the first is multiplied by 2, which is our common ratio \(r\). The series starts with 2, hence \(2\) is the first term \(a\). Summation notation is a concise way to express the entire series: \(\sum_{i=1}^{n} a r^{i-1}\), which, when applied to our example becomes \(\sum_{i=1}^{11} 2^{i}\) by simplifying the formula. This notation leads to a tidy expression, encapsulating the expansive series.

Exponents and Powers

Exponents and powers are shorthand for expressing repeated multiplication. The exponent indicates how many times the base, which is the number being multiplied, is used as a factor. For example, \(2^{3}\) is equivalent to \(2 \cdot 2 \cdot 2\), or \(8\).

In the geometric series given in the exercise, powers of 2 form the backbone of the sequence. The concept of exponents is crucial to comprehend how the series develops over time. Multiplying a number by itself repeatedly can lead to massive values quite rapidly—a key characteristic of exponential growth. It’s the recognition of the underlying pattern of exponents that allows one to transform a long series into a more manageable summation notation.

Sequences and Series

Sequences and series are foundational concepts in mathematics that appear frequently in various branches of the subject, from algebra to calculus. A sequence is an ordered list of numbers following a certain rule, while a series is the sum of a sequence of terms. In our problem, we are looking at a finite series, where the sequence has a definite number of terms.

The power lies in understanding how to convey a long-winded string of numbers as a succinct equation or notation. The summation notation, for example, simplifies the representation of the series and provides a way to calculate the sum systematically. Sequences aren't always as straightforward as a geometric progression, but the approach to summarizing them remains similar—find the underlying rule or pattern and express it mathematically. The earlier problem illustrates a clear application of identifying a geometric sequence and representing it as a geometric series.