Problem 45 - College Algebra

Question

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

Step-by-Step Solution

Answer

The summation notation of the given series is \( \Sigma_{i=1}^{11} {2^{i}} \)

Step 1: Identify the properties of the given series

Note that the series follows a consistent pattern where each term is multiplied by 2, so the common ratio \('r'\) is 2. The first exponent in this series is 1, and the last is 11, which will become the lower and upper bounds of the summation respectively.

Step 2: Write the summation

Substitute these details into the summation formula mentioned above: \( \Sigma_{i=1}^{11} {2^{i}} \)

Key Concepts

Series Representation

A series representation is a way to express a sequence of numbers using a formula that defines the pattern of the numbers. Series can be finite or infinite, but in this case, we have a finite geometric series. A geometric series is characterized by each term being a constant multiple (called the common ratio) of the previous term.
To represent a series using summation notation, we need to identify the pattern in which the terms grow. For example, in the series given in the exercise, each term is formed by multiplying the previous term by 2. This makes it a geometric series with a common ratio of 2.
Understanding the series pattern allows us to condense a lengthy expression into a compact form using mathematical notation, which is both convenient and powerful for calculations.

Exponential Function

The exponential function is a mathematical function where a constant base is raised to a variable exponent. In the provided series, the base 2 is raised to increasing powers starting from 1 up to 11. The function can be expressed as \( 2^i \), where the base 2 is the number repeated in multiplication, and \( i \) is the exponent.
Exponential functions grow rapidly as the value of the exponent increases. They are utilized to model growth processes like population, investment growth, and radioactive decay, as well as in this series for simplifying how we express compounded values.
By using the power of exponents, the series is simplified, displaying the consistent doubling that defines geometric growth in a neat, understandable form.

Mathematical Notation

Mathematical notation is a symbolic language used to express mathematical ideas and formulas succinctly and precisely. It allows mathematicians and students alike to communicate complex concepts efficiently. For series, the Sigma (\( \Sigma \)) notation is widely used to denote the sum of a sequence.
In the problem, \( \Sigma_{i=1}^{11} 2^i \) is the mathematical notation used. It clearly communicates that the terms from \( 2^1 \) through \( 2^{11} \) are summed. The lower and upper bounds are indicated by the numbers under and above the sigma symbol. This form avoids writing out long repetitive sums and aligns with how computers process these calculations.
Being able to read and write in mathematical notation is crucial for tackling more complex mathematical problems efficiently.

Index of Summation

The index of summation is a critical part of the summation notation, represented by the variable in the sequence, often \( i \) or \( n \). It varies from the lower limit to the upper limit of the summation.
In the provided series, \( i \) is the index of summation that helps you to understand that each term of the series is \( 2^i \). Starting at 1 and ending at 11, \( i \) takes each integer value in this range, substituting into the base formula \( 2^i \).
This index helps break down the summation into its component parts, enabling you to understand every individual term in the series. Grasping the concept of the index of summation is paramount in using summation notation to represent series effectively.