Evaluate the Summation sum from i=4 to 12 of -6-5i
The given problem is requesting to calculate the total sum of a particular arithmetic sequence. The sequence starts with i=4 and ends at i=12, and the general term of the sequence is given by the expression -6-5i. The task is to find the sum of all terms of the sequence while incrementing the value of i from 4 up to and including 12.
$\sum_{i = 4}^{12} - 6 - 5 i$
Answer
Solution:
Step 1:
Transform the initial summation to start from $i = 1$ by subtracting the unwanted terms.
$$\sum_{i=4}^{12}(-6-5i) = \sum_{i=1}^{12}(-6-5i) - \sum_{i=1}^{3}(-6-5i)$$
Step 2:
Calculate the summation for $\sum_{i=1}^{12}(-6-5i)$.
Step 2.1:
Decompose the summation into two separate summations.
$$\sum_{i=1}^{12}(-6-5i) = \sum_{i=1}^{12}(-6) + \sum_{i=1}^{12}(-5i)$$
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 given values into the formula.
$$(-6)(12)$$
Step 2.2.3:
Perform the multiplication of $-6$ and $12$.
$$-72$$
Step 2.3:
Compute $-5\sum_{i=1}^{12}i$.
Step 2.3.1:
Apply the arithmetic series summation formula.
$$\sum_{i=1}^{n}i = \frac{n(n+1)}{2}$$
Step 2.3.2:
Place the values into the formula and include the coefficient $-5$.
$$-5\left(\frac{12(12+1)}{2}\right)$$
Step 2.3.3:
Simplify the expression.
Step 2.3.3.1:
Add $12$ and $1$ together.
$$-5\frac{12 \cdot 13}{2}$$
Step 2.3.3.2:
Multiply $12$ by $13$.
$$-5\left(\frac{156}{2}\right)$$
Step 2.3.3.3:
Divide $156$ by $2$.
$$-5 \cdot 78$$
Step 2.3.3.4:
Multiply $-5$ by $78$.
$$-390$$
Step 2.4:
Combine the results of both summations.
$$-72 - 390$$
Step 2.5:
Subtract $390$ from $-72$.
$$-462$$
Step 3:
Determine the summation for $\sum_{i=1}^{3}(-6-5i)$.
Step 3.1:
Write out the terms for each $i$ value.
$$-6 - 5 \cdot 1 - 6 - 5 \cdot 2 - 6 - 5 \cdot 3$$
Step 3.2:
Simplify the series.
Step 3.2.1:
Multiply $-5$ by $1$.
$$-6 - 5 - 6 - 5 \cdot 2 - 6 - 5 \cdot 3$$
Step 3.2.2:
Combine $-5$ and $-6$.
$$-11 - 6 - 5 \cdot 2 - 6 - 5 \cdot 3$$
Step 3.2.3:
Multiply $-5$ by $2$.
$$-11 - 6 - 10 - 6 - 5 \cdot 3$$
Step 3.2.4:
Combine $-10$ and $-6$.
$$-11 - 16 - 6 - 5 \cdot 3$$
Step 3.2.5:
Combine $-16$ and $-11$.
$$-27 - 6 - 5 \cdot 3$$
Step 3.2.6:
Multiply $-5$ by $3$.
$$-27 - 6 - 15$$
Step 3.2.7:
Combine $-15$ and $-6$.
$$-27 - 21$$
Step 3.2.8:
Combine $-21$ and $-27$.
$$-48$$
Step 4:
Substitute the calculated summation values.
$$-462 + 48$$
Step 5:
Add $-462$ and $48$ together.
$$-414$$
Solution:"The sum of the series from i=4 to 12 of -6-5i is -414."
Knowledge Notes:
Summation Notation: Summation notation is a way to represent the sum of a series of terms. It is denoted by the Greek letter sigma ($\Sigma$) and includes the starting index, the ending index, and the expression to sum.
Arithmetic Series: An arithmetic series is the sum of terms in an arithmetic sequence, where each term after the first is obtained by adding a constant difference to the previous term.
Summation of a Constant: The sum of a constant $c$ over $n$ terms is given by $cn$.
Summation of the First $n$ Natural Numbers: The sum of the first $n$ natural numbers is given by $\frac{n(n+1)}{2}$.
Decomposition of a Summation: A summation of a series with multiple terms can be broken down into the sum of individual summations for each term.
Simplification: Simplification involves performing arithmetic operations such as addition, subtraction, multiplication, and division to reduce expressions to a simpler form.
