Problem 46 - Algebra and Trigonometry

Question

Find the sum. $$\sum_{i=1}^{3} i 2^{i}$$

Step-by-Step Solution

Answer

The sum is 34.

Step 1: Understand the Summation Notation

The expression \( \sum_{i=1}^{3} i 2^{i} \) is a summation where \( i \) is the index of summation, \( 2^{i} \) is an exponential function, and the sum is taken from \( i = 1 \) to \( i = 3 \).

Step 2: Expand the Summation

Expand the summation by explicitly writing out the terms. This will give you:\[1 \cdot 2^{1} + 2 \cdot 2^{2} + 3 \cdot 2^{3}\]

Step 3: Compute Each Term

Calculate each term in the expanded summation:- For \( i = 1 \): \( 1 \cdot 2^{1} = 2 \)- For \( i = 2 \): \( 2 \cdot 2^{2} = 8 \)- For \( i = 3 \): \( 3 \cdot 2^{3} = 24 \)

Step 4: Add the Terms Together

Add the results from Step 3:\[2 + 8 + 24 = 34\]

Step 5: Verify the Answer

Check your calculations:- The calculations of \( 1 \cdot 2^{1}, 2 \cdot 2^{2}, \) and \( 3 \cdot 2^{3} \) are correct.- The addition \( 2 + 8 + 24 \) correctly gives \( 34 \).Thus, the answer is verified.

Key Concepts

Understanding the Index of Summation

In summation notation, the index of summation plays a critical role. It specifies the variable as it changes through a range of values. It is vital to understand how this works to solve a summation problem.
In the given expression \( \sum_{i=1}^{3} i 2^{i} \), \( i \) is the index of summation. It indicates that we begin at \( i = 1 \) and continue adding terms until \( i = 3 \).
This variable methodically moves through the sequence of numbers, ensuring each term is accounted for in the final summation.

• The starting value is indicated below the summation symbol \( \Sigma \), which in our example is 1.
• The ending value is above the summation symbol, which is 3 in this exercise.

By carefully following the index of summation, the structured process of addition is maintained, ensuring no term is overlooked.

Exploring the Exponential Function

Exponential functions are a key component in mathematics, where their rate of change grows exponentially with respect to the variable. In our summation problem, the term \( 2^{i} \) represents an exponential function, where 2 is the base and \( i \) is the exponent.
As the index \( i \) increases, the power of 2 multiplies by itself, quickly increasing the size of the result.
When computing the sum, each term incorporates a different power of 2:

• For \( i = 1 \), we have \( 2^{1} = 2 \).
• For \( i = 2 \), we see \( 2^{2} = 4 \).
• For \( i = 3 \), it becomes \( 2^{3} = 8 \).

This rapid escalation highlights the potency of exponential functions, especially when seen over several sequential terms as in this summation.

The Importance of Mathematical Verification

Verification is a vital step in solving mathematical problems to ensure that the solution is accurate and reliable. Once all initial calculations are conducted, thorough checking solidifies the result.
For our problem, after calculating each term in the summation individually, it's important to verify:

• First, we recheck the multiplication for each term: \( 1 \cdot 2^{1}, 2 \cdot 2^{2}, \) and \( 3 \cdot 2^{3} \).
• Secondly, we confirm the addition process: \( 2 + 8 + 24 \).

This careful re-evaluation ensures each part of the process is correct and identifies any possible mistakes.
Mathematical verification elevates the problem-solving process from simply finding an answer to ensuring it's conclusively correct, guaranteeing the integrity of the result. It is an indispensable part of any mathematical procedure.