Evaluate the Summation sum from i=8 to 6 of (i-1)^2
The problem is asking for the calculation of a finite mathematical series. It starts by establishing the index of summation, "i", and the range over which it should be executed, which in this case is somewhat unusual as it seems to suggest counting backwards, starting from i=8 and ending at i=6. For each value of i within this range, you are instructed to take that value, subtract 1 from it, square the result, and then sum all those squared values together. However, the traditional interpretation of this notation would imply that the summation is empty (since normally the index would start from a smaller value and increase to a larger one), so it's likely there's been a typographical error in the question or it's intentionally formulated this way to test understanding of summation notation. The request is to only explain what the problem is asking for and not to actually perform the calculation.
$\sum_{i = 8}^{6} \left(\left(\right. i - 1 \left.\right)\right)^{2}$
Answer
Solution:
Step 1: Simplify the given summation expression.
Step 1.1: Express $(i - 1)^2$ as $(i - 1)(i - 1)$.
Step 1.2: Expand the expression $(i - 1)(i - 1)$ by applying the FOIL (First, Outer, Inner, Last) method.
Step 1.2.1: Distribute $i$ over $(i - 1)$ to get $i(i - 1)$ and then distribute $-1$ over $(i - 1)$ to get $-1(i - 1)$.
Step 1.2.2: Continue distributing to obtain $i \cdot i - i - 1 \cdot i + 1$.
Step 1.2.3: Finalize distribution to get $i^2 - i - i + 1$.
Step 1.3: Combine like terms and simplify the expression.
Step 1.3.1: Simplify each term individually.
Step 1.3.1.1: $i \cdot i$ simplifies to $i^2$.
Step 1.3.1.2: $-1 \cdot i$ simplifies to $-i$.
Step 1.3.1.3: Rewrite $-i$ as is.
Step 1.3.1.4: Rewrite $-i$ as is.
Step 1.3.1.5: $-1 \cdot -1$ simplifies to $+1$.
Step 1.3.2: Combine the terms to get $i^2 - 2i + 1$.
Step 1.4: Represent the simplified expression in summation notation as $\sum_{i=8}^{6} (i^2 - 2i + 1)$.
Step 2: Evaluate the summation.
- Since the lower limit of summation (8) is greater than the upper limit (6), the summation is empty and therefore the sum is 0.
Knowledge Notes:
The problem involves evaluating a summation with an upper and lower limit. The summation notation $\sum$ is used to denote the sum of a sequence of terms defined by an expression inside the summation. The process of evaluating a summation includes simplifying the expression inside the summation and then calculating the sum of the terms within the limits provided.
Simplifying the Expression: Before evaluating the sum, the expression inside the summation needs to be simplified. This often involves expanding and simplifying algebraic expressions, which in this case is $(i - 1)^2$.
FOIL Method: The FOIL method stands for First, Outer, Inner, Last, which is a technique used to expand binomials. When two binomials are multiplied, each term in the first binomial is multiplied by each term in the second binomial, and the results are added together.
Distributive Property: This property is used to multiply a single term by each term in a binomial or polynomial. It is often used in conjunction with the FOIL method to expand expressions.
Combining Like Terms: After expansion, terms that are alike (i.e., have the same variables raised to the same power) are combined to simplify the expression further.
Empty Summation: A summation is considered empty if the lower limit is greater than the upper limit. By convention, the value of an empty summation is zero.
Summation Notation: The summation notation includes the expression to be summed, the index of summation (in this case, $i$), and the upper and lower limits of summation. For example, $\sum_{i=a}^{b} f(i)$ means to sum the function $f(i)$ for all integer values of $i$ from $a$ to $b$ inclusive.
In this problem, the summation is empty because the lower limit (8) is greater than the upper limit (6), so the sum is defined to be zero without needing to calculate the individual terms.
