Math Problem Statement

Write the sum 5 + 4 + 3 + ... + (-7) using sigma notation.

Solution

The sum provided in the image is:
5+4+3+2+1+0+(1)+(2)+(3)+(4)+(5)+(6)+(7)5 + 4 + 3 + 2 + 1 + 0 + (-1) + (-2) + (-3) + (-4) + (-5) + (-6) + (-7)

We need to express this sum using sigma notation. To break this down:

  1. The first term is 5, and the last term is -7.
  2. The terms decrease by 1 for each successive term, starting at 5 and ending at -7.

Thus, we can express this as:

i=113(6i)\sum_{i=1}^{13} (6 - i)

Here’s the breakdown:

  • i=1i = 1 gives 61=56 - 1 = 5,
  • i=2i = 2 gives 62=46 - 2 = 4,
  • and so on, until i=13i = 13, which gives 613=76 - 13 = -7.

This sum is written in sigma notation as: i=113(6i)\sum_{i=1}^{13} (6 - i)

Let me know if you need further clarifications or help with another concept!


Related questions for you:

  1. What is the sum of this series if you compute it manually?
  2. How would you express this sum if the terms were increasing by 2 instead of decreasing by 1?
  3. What is the formula for the sum of an arithmetic series?
  4. Can you write the sum of all even numbers between 1 and 20 using sigma notation?
  5. What changes to the formula would be needed if we started at a different number than 6?

Tip: When writing a sum in sigma notation, identify the pattern in the terms and express it as a function of the summation index.

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Math Problem Analysis

Mathematical Concepts

Sigma Notation
Arithmetic Sequences

Formulas

Σ notation for summation
Arithmetic sequence formula

Theorems

Sum of an arithmetic series

Suitable Grade Level

Grades 7-9