Evaluate the Summation sum from i=1 to 6 of 2-i-i^2
The question asks to calculate the sum of a series where the terms are given by a specific formula, which is 2 - i - i^2, and the summation is to be carried out from i=1 to i=6. In this summation, i represents each integer from 1 through 6, and for each of these values of i, the expression 2 - i - i^2 is computed and all the results are then added together to obtain the final sum.
$\sum_{i = 1}^{6} 2 - i - i^{2}$
Answer
Solution:
Step 1: Decompose the given summation into individual summations.
$\sum_{i = 1}^{6} (2 - i - i^2) = \sum_{i = 1}^{6} 2 - \sum_{i = 1}^{6} i - \sum_{i = 1}^{6} i^2$
Step 2: Calculate the summation of the constant term.
$\sum_{i = 1}^{6} 2$
Step 2.1: Apply the formula for the sum of a constant.
$\sum_{i = 1}^{n} c = cn$
Step 2.2: Insert the given values into the formula.
$(2)(6)$
Step 2.3: Perform the multiplication.
$2 \times 6 = 12$
Step 3: Calculate the summation of the first-degree term.
$\sum_{i = 1}^{6} i$
Step 3.1: Use the formula for the sum of the first n natural numbers.
$\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$
Step 3.2: Plug in the numbers and include the negative sign.
$(-1) \left(\frac{6(6 + 1)}{2}\right)$
Step 3.3: Simplify the expression.
Step 3.3.1: Add the numbers inside the parentheses.
$-1 \frac{6 \times 7}{2}$
Step 3.3.2: Multiply the numbers.
$-1 \left(\frac{42}{2}\right)$
Step 3.3.3: Divide the result.
$-1 \times 21$
Step 3.3.4: Multiply by the negative sign.
$-21$
Step 4: Calculate the summation of the second-degree term.
$\sum_{i = 1}^{6} i^2$
Step 4.1: Use the formula for the sum of the squares of the first n natural numbers.
$\sum_{i = 1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6}$
Step 4.2: Insert the values into the formula and account for the negative sign.
$(-1) \left(\frac{6(6 + 1)(2 \times 6 + 1)}{6}\right)$
Step 4.3: Simplify the expression.
Step 4.3.1: Work on the numerator.
Step 4.3.1.1: Add the numbers.
$-1 \frac{6 \times 7 (2 \times 6 + 1)}{6}$
Step 4.3.1.2: Multiply the numbers.
$-1 \frac{42(12 + 1)}{6}$
Step 4.3.1.3: Add the numbers.
$-1 \frac{42 \times 13}{6}$
Step 4.3.2: Simplify the fraction.
Step 4.3.2.1: Multiply the numbers.
$-1 \left(\frac{546}{6}\right)$
Step 4.3.2.2: Divide the result.
$-1 \times 91$
Step 4.3.2.3: Multiply by the negative sign.
$-91$
Step 5: Combine the results of the individual summations.
$12 - 21 - 91$
Step 6: Simplify the final result.
Step 6.1: Subtract the second term from the first.
$-9 - 91$
Step 6.2: Subtract the third term from the result.
$-100$
Knowledge Notes:
To solve the given problem, we need to understand the following concepts:
Summation Notation: Summation notation is a convenient way to represent the sum of a series of terms. It is denoted by the Greek letter sigma ($\Sigma$) and includes an expression for the terms to be summed, along with the index of summation and its range.
Sum of a Constant: The sum of a constant $c$ over $n$ terms is simply $cn$. This is because each term in the summation is the same, and there are $n$ such terms.
Sum of the First n Natural Numbers: The sum of the first $n$ natural numbers is given by the formula $\frac{n(n + 1)}{2}$.
Sum of the Squares of the First n Natural Numbers: The sum of the squares of the first $n$ natural numbers is given by the formula $\frac{n(n + 1)(2n + 1)}{6}$.
Decomposition of Summations: A summation of multiple terms can be broken down into separate summations of each term. This is useful when the individual terms have known summation formulas.
Arithmetic Operations: Basic arithmetic operations like addition, subtraction, multiplication, and division are used to simplify the expressions obtained from the summation formulas.
By applying these concepts and formulas, we can evaluate the given summation step by step, breaking it down into simpler parts and using known formulas to find the sum of each part.
