Sigma notation Evaluate the following expressions.
(d)     5                                                                                                                                                                              
       ∑ (1 + n²)                                                                                                                                                                          
       n=1                         

Verified step by step guidance

1. Step 1: Understand the problem. The given expression is a summation in sigma notation: ∑ (1 + n²) from n = 1 to 5. This means we need to calculate the sum of the terms (1 + n²) for each integer value of n starting at 1 and ending at 5.
2. Step 2: Write out the individual terms of the summation. Substitute each value of n (from 1 to 5) into the expression (1 + n²). The terms will be: (1 + 1²), (1 + 2²), (1 + 3²), (1 + 4²), and (1 + 5²).
3. Step 3: Simplify each term. For example, (1 + 1²) simplifies to 2, (1 + 2²) simplifies to 5, and so on. Perform this simplification for all terms.
4. Step 4: Add the simplified terms together. Once all terms are simplified, sum them up to find the total value of the summation.
5. Step 5: Verify your work. Double-check each substitution, simplification, and addition to ensure accuracy in your calculations.

Key Concepts

Sigma Notation

Sigma notation is a concise way to represent the sum of a sequence of terms. It uses the Greek letter sigma (Σ) to indicate summation, followed by an expression that defines the terms to be summed. The notation includes limits that specify the starting and ending indices of the summation, allowing for efficient calculation of series.

Series and Sequences

A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence. Understanding the difference is crucial for evaluating expressions in sigma notation. In the context of the given question, the series involves summing the values of the expression (1 + n²) for each integer n within specified limits.

Evaluating Summations

Evaluating summations involves calculating the total of a series of numbers generated by a specific formula. This requires substituting the index values into the expression and performing the arithmetic operations. Mastery of this concept is essential for accurately solving problems that involve sigma notation, such as the one presented in the question.