Sigma Notation: Understanding the Formula 36 = Σn (1/2n(n+1))
What is Sigma Notation?
Sigma notation, also known as summation notation, is a mathematical notation used to represent the sum of a finite number of terms. It is denoted by the Greek letter Σ (sigma). The notation is used to simplify the representation of a sum of multiple terms, making it easier to write and calculate.
The Formula: 36 = Σn (1/2n(n+1))
Let's break down the formula: 36 = Σn (1/2n(n+1))
In this formula, Σ is the sigma notation, n is the variable of summation, and 1/2n(n+1) is the formula being summed.
To understand this formula, let's expand it:
36 = (1/2 × 1 × 2) + (1/2 × 2 × 3) + (1/2 × 3 × 4) + ... + (1/2 × n × (n+1))
Simplifying the Formula
To simplify the formula, we can start by multiplying the numerator and denominator of each term:
36 = (1/2) [1 × 2 + 2 × 3 + 3 × 4 + ... + n × (n+1)]
Now, let's focus on the sum:
1 × 2 + 2 × 3 + 3 × 4 + ... + n × (n+1)
This sum can be rewritten as:
2 + 6 + 12 + ... + n(n+1)
The Pattern: Triangular Numbers
The sum of the first n triangular numbers is equal to n(n+1)/2. This is a well-known mathematical formula.
Using this formula, we can rewrite the sum:
2 + 6 + 12 + ... + n(n+1) = n(n+1)/2
Proving the Formula
Now, let's prove that the formula 36 = Σn (1/2n(n+1)) is true.
36 = n(n+1)/2
To find the value of n, we can solve the equation:
36 = n(n+1)/2
Multiplying both sides by 2, we get:
72 = n(n+1)
Expanding the right-hand side, we get:
72 = n^2 + n
Rearranging the equation, we get:
n^2 + n - 72 = 0
Factoring the quadratic equation, we get:
(n - 6)(n + 12) = 0
This gives us two possible values for n: n = 6 and n = -12.
Since n cannot be negative, we take n = 6.
Substituting n = 6 back into the formula, we get:
36 = 6(6+1)/2
Simplifying, we get:
36 = 36
Therefore, the formula 36 = Σn (1/2n(n+1)) is true when n = 6.
Conclusion
In this article, we explored the sigma notation formula 36 = Σn (1/2n(n+1)). We broke down the formula, expanded it, and simplified it. We also proved that the formula is true when n = 6. Understanding sigma notation and its applications can help us solve complex mathematical problems and appreciate the beauty of mathematics.
