Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

\(\sum_{k=0}^{4} 3\left(2^{k}\right)\)

Step-by-Step Solution

Answer

The sum is 93.

Step 1: Understand the Expression

The given exercise is a summation notation of the form \(\sum_{k=0}^{4} 3\left(2^{k}\right)\). It means that we need to calculate the sum of the terms when \(k\) takes values from 0 to 4 inclusive. Each term is in the form \(3(2^k)\).

Step 2: Expand the Summation

We will expand the summation \(\sum_{k=0}^{4} 3\left(2^{k}\right)\) by substituting values for \(k\) from 0 to 4.- When \(k = 0\), the term is \(3 \times 2^0 = 3 \times 1 = 3\).- When \(k = 1\), the term is \(3 \times 2^1 = 3 \times 2 = 6\).- When \(k = 2\), the term is \(3 \times 2^2 = 3 \times 4 = 12\).- When \(k = 3\), the term is \(3 \times 2^3 = 3 \times 8 = 24\).- When \(k = 4\), the term is \(3 \times 2^4 = 3 \times 16 = 48\).

Step 3: Sum the Expanded Terms

Now, sum up all the calculated terms from the expanded expression: \[ 3 + 6 + 12 + 24 + 48 \]

Step 4: Calculate the Total

To find the total, simply add the numbers together:\[ 3 + 6 + 12 + 24 + 48 = 93 \]

Key Concepts

Understanding Arithmetic Sequences

Arithmetic sequences are a series of numbers where the difference between consecutive terms remains constant. This difference is called the "common difference." For example, the sequence 2, 5, 8, 11 has a common difference of 3, because each term increases by 3. You can generalize an arithmetic sequence using the formula: \[ a_n = a_1 + (n-1)d \]where:

• \( a_n \) is the \( n \)-th term of the sequence
• \( a_1 \) is the first term
• \( d \) is the common difference
• \( n \) is the term number

This simple structure makes it easy to predict any term in the sequence, as long as you know the first term and the common difference.
In relation to the given exercise, although the terms increase as they would in an arithmetic sequence, each term specifically involves exponential calculations rather than simple additions.

What are Exponential Functions?

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. For example, in \( f(x) = a^x \), \( a \) is the base, and \( x \) is the exponent. Exponential growth or decay is commonly observed in nature, such as in population growth or radioactive decay.
Some key properties of exponential functions include:

• Exponential growth rates increase rapidly.
• The function \( a^x \) is continuous and only positive if \( a > 0 \).
• \( a^0 = 1 \) for any non-zero \( a \).

In the given exercise, the exponential nature of powers is shown in \( 2^k \). By multiplying it with constants (in this case, 3), the exercise explores a sequence of numbers which grows exponentially rather than linearly.

Using Summation Notation

Summation notation is a concise way of expressing the sum of a sequence of terms. This uses the Greek letter \( \Sigma \), standing for "sum." You might see it written as \( \sum_{i=m}^{n} f(i) \), which means to sum up \( f(i) \) as \( i \) goes from \( m \) to \( n \).
Here's how to break it down:

• \( m \) is the starting index, where the summation begins.
• \( n \) is the ending index, where the summation stops.
• \( f(i) \) is the function to be summed, often dependent on the index \( i \).

This notation is beneficial because it simplifies the representation of long sums. For instance, the exercise \( \sum_{k=0}^{4} 3\left(2^{k}\right) \) directs you to perform the sum of terms like \( 3(2^0) \) up to \( 3(2^4) \). Each term is derived from substituting \( k \) within the bounds of 0 to 4 in the expression \( 3(2^k) \). This method saves time and space, especially when working with complex or lengthy sequences.