Section 2.2 – Sigma and Subscript Notation

• Definition of Sigma Notation

  • Sigma (Σ) notation is a concise way of expressing the sum of a sequence of numbers.

• Notation Breakdown

  • General form: $ext{Sum}(i=1 ext{ to } n) ext{ of } x<em>i = ext{Σ}</em>{i=1}^{n} x_i$
  • Here, $x1$, $x2$, …, $x_n$ represent the terms of the sequence.
  • The variable $i$ is called the index of summation and is used to iterate through the values.
  • The limits of summation, from $i=1$ to $n$, indicate that the sum starts at $i = 1$ and ends at $i = n$.

• Example of Summation

  • For a specific sequence of numbers:
  • $ext{Σ}<em>{i=1}^{n} x</em>i = x<em>1 + x</em>2 + ext{…} + x_n$

• Summation of Squares

  • The notation for the sum of the squares of the terms is represented as:
  • $ext{Σ}<em>{i=1}^{n} x</em>i^2 = x<em>1^2 + x</em>2^2 + ext{…} + x_n^2$

Example Calculations

• Example 1: Simple Sum

  • Given a set of values: 4, 3, 1, 4
  • Calculate total using summation notation:
    • $ext{Σ}<em>{i=1}^{n} x</em>i$ gives
    • $ext{Σ}<em>{i=1}^{4} x</em>i = 4 + 3 + 1 + 4$
    • Total = 12.

• Summation of Squares of Example 1

  • Using the previously defined square summation notation:
  • $12^2 = ext{Σ}<em>{i=1}^{n} x</em>i^2 = 4^2 + 3^2 + 1^2 + 4^2$
  • Calculation Steps:
    • $= 16 + 9 + 1 + 16$
    • Total = 42
  • Note:
    • $ext{Σ}<em>{i=1}^{n} x</em>i^2 ext{ results in } 42$
    • Conclusion:
      • $42 <br />\neq 12^2$
      • This example illustrates the difference between the square of a sum and the sum of squares.

Additional Example Calculations

• Example 2: Weighted Sum

  • Utilize the notation for a weighted sum involving pairs of $xi$ and $yi$:
  • $ext{Σ}<em>{i=1}^{n} x</em>i y_i$
    • Given a specific calculation:
    • Totaling to 59
    • Example calculation with pairs:
    • $= (4)(5) + (3)(4) + (1)(3) + (4)(6)$
    • This equals 59.

• Calculating Individual Summations

  • Also compute the separate sums of each sequence:
  • $ext{Σ}<em>{i=1}^{n} x</em>i = 12$
  • $ext{Σ}<em>{i=1}^{n} y</em>i = 18$
  • From these, calculate combined weighted sum:
    • $ext{Σ} x<em>i ext{Σ} y</em>i = 12 imes 18$
    • Total = 216
  • Note:
  • The result shows:
    • $x<em>i ext{ and } y</em>i ext{ pairs do not equal the weighted sum formed originally.}$