Sigma notation Evaluate the following expressions.
(a)    10                                                                                                                                                                               
       ∑ κ                                                                                                                                                                          
       κ=1                         

Verified step by step guidance

1. Step 1: Understand the problem. The given expression involves sigma notation, which represents the summation of terms. The general form of sigma notation is ∑_{k=a}^{b} f(k), where 'k' is the index of summation, 'a' is the lower limit, 'b' is the upper limit, and f(k) is the function to be summed.
2. Step 2: Identify the components of the given sigma notation. In this case, the summation is ∑_{k=1}^{10} k, which means we are summing the values of 'k' from 1 to 10.
3. Step 3: Write out the terms of the summation explicitly. Substitute the values of 'k' from 1 to 10 into the expression 'k'. This gives: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
4. Step 4: Recognize that this is an arithmetic series. The sum of the first 'n' natural numbers can be calculated using the formula S = n(n+1)/2, where 'n' is the largest number in the series.
5. Step 5: Apply the formula for the sum of the first 'n' natural numbers. Substitute n = 10 into the formula S = n(n+1)/2 to find the sum of the series. This will give the final result.

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 typically includes limits that specify the starting and ending indices of the summation, allowing for efficient representation of large sums.

Index of Summation

The index of summation is a variable that represents the position of each term in the sequence being summed. It is usually denoted by a letter, such as 'k', and takes on integer values from a specified lower limit to an upper limit. Understanding how to manipulate and evaluate the index is crucial for correctly calculating the sum represented by sigma notation.

Evaluating Series

Evaluating series involves calculating the total sum of the terms defined by the sigma notation. This process may require substituting values for the index of summation, performing arithmetic operations, and sometimes applying formulas for known series. Mastery of techniques for evaluating series is essential for solving problems that involve sigma notation.