Evaluate the Summation sum from i=4 to 12 of -8-7i
The given problem is about calculating the result of a finite arithmetic series. The summation indicated is asking for the sum of terms from the 4th term to the 12th term, each term being defined by the arithmetic expression -8-7i, where i is the index of the term within the series. You are expected to find the cumulative sum of this sequence for each value of i from 4 to 12.
$\sum_{i = 4}^{12} - 8 - 7 i$
Answer
Solution:
Step 1:
Decompose the given summation to start from $i=1$.
$$\sum_{i=4}^{12} (-8-7i) = \sum_{i=1}^{12} (-8-7i) - \sum_{i=1}^{3} (-8-7i)$$
Step 2:
Calculate $\sum_{i=1}^{12} (-8-7i)$.
Step 2.1:
Separate the summation into two parts.
$$\sum_{i=1}^{12} (-8-7i) = \sum_{i=1}^{12} (-8) - 7\sum_{i=1}^{12} i$$
Step 2.2:
Compute $\sum_{i=1}^{12} (-8)$.
Step 2.2.1:
Use the constant summation rule.
$$\sum_{i=1}^{n} c = cn$$
Step 2.2.2:
Insert the given numbers into the formula.
$$(-8)(12)$$
Step 2.2.3:
Perform the multiplication.
$$-96$$
Step 2.3:
Find $7\sum_{i=1}^{12} i$.
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 and include 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 $1$.
$$-7\frac{12 \cdot 13}{2}$$
Step 2.3.3.2:
Calculate $12 \times 13$.
$$-7\left(\frac{156}{2}\right)$$
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 both parts.
$$-96 - 546$$
Step 2.5:
Subtract $546$ from $-96$.
$$-642$$
Step 3:
Compute $\sum_{i=1}^{3} (-8-7i)$.
Step 3.1:
Expand the series for each value of $i$.
$$-8 - 7 \cdot 1 - 8 - 7 \cdot 2 - 8 - 7 \cdot 3$$
Step 3.2:
Simplify the series.
Step 3.2.1:
Multiply $-7$ by $1$.
$$-8 - 7 - 8 - 7 \cdot 2 - 8 - 7 \cdot 3$$
Step 3.2.2:
Combine $-7$ with $-8$.
$$-15 - 8 - 7 \cdot 2 - 8 - 7 \cdot 3$$
Step 3.2.3:
Multiply $-7$ by $2$.
$$-15 - 8 - 14 - 8 - 7 \cdot 3$$
Step 3.2.4:
Combine $-14$ with $-8$.
$$-15 - 22 - 8 - 7 \cdot 3$$
Step 3.2.5:
Combine $-22$ with $-15$.
$$-37 - 8 - 7 \cdot 3$$
Step 3.2.6:
Multiply $-7$ by $3$.
$$-37 - 8 - 21$$
Step 3.2.7:
Combine $-21$ with $-8$.
$$-37 - 29$$
Step 3.2.8:
Combine $-29$ with $-37$.
$$-66$$
Step 4:
Substitute the summation values found.
$$-642 + 66$$
Step 5:
Add $-642$ and $66$.
$$-576$$
Knowledge Notes:
Summation notation $\sum$ is a way to represent the addition of a sequence of numbers. The variable $i$ is the index of summation, and the numbers at the top and bottom of the $\sum$ symbol indicate the range of indices to sum over.
When summing a constant $c$ over $n$ terms, the result is simply $cn$.
The summation of the first $n$ natural numbers is given by the formula $\frac{n(n+1)}{2}$.
Arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term.
When dealing with summations, it is often useful to separate the summation into simpler parts that can be individually calculated and then recombined.
In this problem, we used the properties of summations to split the original summation into two parts, one involving a constant and the other involving a linear term. We then applied the appropriate formulas to each part before recombining them to get the final result.
