Evaluate the Summation sum from i=4 to 12 of -6-7i
The problem is a mathematical exercise in evaluating a summation. Summations are mathematical expressions that denote the addition of a sequence of numbers. In this particular problem, you are asked to calculate the sum of a linear expression, "-6-7i," for every integer value of the variable "i" from 4 to 12, inclusive. The "-6" is a constant term and "-7i" is a variable term where "i" takes on each integer value within the range specified. The result would be the total sum after adding up the values of this expression for each value of "i" in the given range.
$\sum_{i = 4}^{12} - 6 - 7 i$
Answer
Solution:
Step 1:
Transform the original summation to start from $i=1$.
$\sum_{i=4}^{12} (-6-7i) = \left(\sum_{i=1}^{12} (-6-7i)\right) - \left(\sum_{i=1}^{3} (-6-7i)\right)$
Step 2:
Calculate $\sum_{i=1}^{12} (-6-7i)$.
Step 2.1:
Decompose the summation into two separate summations.
$\sum_{i=1}^{12} (-6-7i) = \sum_{i=1}^{12} (-6) + \sum_{i=1}^{12} (-7i)$
Step 2.2:
Compute $\sum_{i=1}^{12} (-6)$.
Step 2.2.1:
Use the constant summation formula.
$\sum_{i=1}^{n} c = cn$
Step 2.2.2:
Insert the values into the formula.
$(-6)(12)$
Step 2.2.3:
Perform the multiplication of $-6$ and $12$.
$-72$
Step 2.3:
Determine $7\sum_{i=1}^{12} i$.
Step 2.3.1:
Apply the arithmetic series formula.
$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
Step 2.3.2:
Place the values into the formula, including the coefficient.
$(-7) \left(\frac{12(12+1)}{2}\right)$
Step 2.3.3:
Simplify the expression.
Step 2.3.3.1:
Add $12$ and $1$ together.
$-7 \frac{12 \cdot 13}{2}$
Step 2.3.3.2:
Multiply $12$ by $13$.
$-7 \left(\frac{156}{2}\right)$
Step 2.3.3.3:
Divide $156$ by $2$.
$-7 \cdot 78$
Step 2.3.3.4:
Multiply $-7$ by $78$.
$-546$
Step 2.4:
Combine the results of the two summations.
$-72 - 546$
Step 2.5:
Subtract $546$ from $-72$.
$-618$
Step 3:
Evaluate $\sum_{i=1}^{3} (-6-7i)$.
Step 3.1:
Write out the series for each value of $i$.
$(-6 - 7 \cdot 1) + (-6 - 7 \cdot 2) + (-6 - 7 \cdot 3)$
Step 3.2:
Simplify the series.
Step 3.2.1:
Calculate $-7$ times $1$.
$-6 - 7 - 6 - 14 - 6 - 21$
Step 3.2.2:
Combine $-7$ and $-6$.
$-13 - 6 - 14 - 6 - 21$
Step 3.2.3:
Multiply $-7$ by $2$.
$-13 - 20 - 6 - 21$
Step 3.2.4:
Combine $-14$ and $-6$.
$-13 - 20 - 27$
Step 3.2.5:
Combine $-20$ and $-13$.
$-33 - 27$
Step 3.2.6:
Multiply $-7$ by $3$.
$-33 - 27$
Step 3.2.7:
Combine $-21$ and $-6$.
$-33 - 27$
Step 3.2.8:
Combine $-27$ and $-33$.
$-60$
Step 4:
Substitute the calculated summations.
$-618 + 60$
Step 5:
Add $-618$ and $60$ together.
$-558$
Knowledge Notes:
Summation Notation: Summation notation is a way to represent the sum of a series of terms. It is denoted by the symbol $\sum$ and includes an expression to be summed, along with the index of summation and its range.
Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term.
Summation of Constants: The sum of a constant $c$ over $n$ terms is given by the formula $\sum_{i=1}^{n} c = cn$.
Summation of the First $n$ Positive Integers: The sum of the first $n$ positive integers is given by the formula $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.
Decomposition of Summations: A summation of multiple terms can be decomposed into the sum of separate summations if the terms are additive.
Index Shift for Summations: To change the index of summation, one can adjust the limits of summation and compensate by adding or subtracting the terms that are excluded or included due to the shift.
By understanding these concepts, one can effectively manipulate and evaluate summations, which is a fundamental skill in mathematics, especially in calculus and discrete mathematics.
