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Chapter 8: Problem 42

Find the sum. \(\sum_{k=1}^4 3 k^2\)

Short Answer

Expert verified
The sum is 90.

Step by step solution

01

Understand the Sigma Notation & Generate Sequence

The sigma notation stipulates that the formula following the sigma, in this case, \(3k^2\), is to be computed for the values of k from 1 to 4, inclusive. Now generate the sequence by plugging in these values into \(3k^2\). The sequence is \(3*1^2, 3*2^2, 3*3^2, 3*4^2\).
02

Compute Each Term of the Sequence

Compute each term of the series individually: \(3*1^2 = 3, 3*2^2 = 12, 3*3^2 = 27, 3*4^2 = 48\).
03

Add the Values

The sum of these values \(3, 12, 27, 48\) equals to 90.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Summation
Summation, in mathematics, is the process of adding a sequence of numbers together. It is often represented using the Greek letter sigma (\( \Sigma \)), followed by an expression that depends on a variable, such as \(3k^2\) in our example, that is evaluated at specific integer values. The example provided is \[\sum_{k=1}^4 3k^2.\]This notation means that the value of the expression \(3k^2\) should be calculated for each integer \(k\) from 1 to 4, and then all these results are to be added together. Summation is a crucial tool in mathematics, allowing for the concise representation of sums of sequences or series, which might otherwise be quite verbose.
Sequence Generation
Sequence generation is a critical step in working with sigma notation. It involves creating a list of numbers by evaluating the expression given in the sigma notation for each integer in a specified range. For the given problem:
  • First, identify the range, which here is \(k = 1\) to \(k = 4\).
  • Then, substitute these values into the expression \(3k^2\).
  • This results in a sequence: \(3 \times 1^2, 3 \times 2^2, 3 \times 3^2, 3 \times 4^2\).
The values derived from this are the numbers we will sum. Understanding how to systematically generate a sequence from a given formula is fundamental to solving problems involving sequences and series.
Arithmetic Operations
Arithmetic operations involve basic mathematical calculations, namely addition, subtraction, multiplication, and division. In tasks involving summation, arithmetic operations are crucial since they help in calculating the value of each term and eventually the total sum. Breaking it down:
  • Start with evaluating each term: For instance, calculate \(3 \times 1^2 = 3\).
  • Perform similar multiplications for all terms in the sequence: \(3 \times 2^2 = 12, 3 \times 3^2 = 27, 3 \times 4^2 = 48.\)
  • Finally, sum these results: \(3 + 12 + 27 + 48 = 90\).
Mastering arithmetic operations ensures accuracy and efficiency when handling any summation problems.

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