Evaluate the Summation sum from n=1 to 8 of (-1)^n
The question is asking to calculate the value of a finite mathematical series. In the series provided, the summation is to be taken for values of $n$from 1 to 8, with the term to be summed being $(-1)^n$. The term $(-1)^n$represents an alternating sign function, which means it will give 1 when $n$is even and -1 when $n$is odd. The question requires the application of sequence and series concepts to determine the combined value of these eight terms once the alternating pattern is accounted for.
$\sum_{n = 1}^{8} \left(\left(\right. - 1 \left.\right)\right)^{n}$
Answer
Solution:
Step 1:
To calculate the sum of a finite geometric series, use the formula $S = a \left( \frac{1 - r^n}{1 - r} \right)$, where $a$ is the initial term and $r$ is the common ratio.
Step 2:
Determine the common ratio $r$ by using the formula $r = \frac{a_{n+1}}{a_n}$.
Step 2.1:
Insert $a_n$ and $a_{n+1}$ into the ratio formula: $r = \frac{(-1)^{n+1}}{(-1)^n}$.
Step 2.2:
Eliminate the similar factor from $(-1)^{n+1}$ and $(-1)^n$.
Step 2.2.1:
Extract $(-1)^n$ from $(-1)^{n+1}$: $r = \frac{(-1)^n \cdot (-1)}{(-1)^n}$.
Step 2.2.2:
Remove the common factors.
Step 2.2.2.1:
Multiply by unity: $r = \frac{(-1)^n \cdot (-1)}{(-1)^n \cdot 1}$.
Step 2.2.2.2:
Cancel out the common factor: $r = \frac{\cancel{(-1)^n} \cdot (-1)}{\cancel{(-1)^n} \cdot 1}$.
Step 2.2.2.3:
Reformulate the expression: $r = \frac{-1}{1}$.
Step 2.2.2.4:
Divide $-1$ by $1$: $r = -1$.
Step 3:
Identify the first term by substituting the lower limit into $(-1)^n$.
Step 3.1:
Replace $n$ with $1$: $a = (-1)^1$.
Step 3.2:
Compute the exponent: $a = -1$.
Step 4:
Plug in the values for $r$, $a$, and the number of terms into the summation formula: $S = - \frac{1 - (-1)^8}{1 - (-1)}$.
Step 5:
Simplify the expression.
Step 5.1:
Condense the numerator.
Step 5.1.1:
Combine $-1$ with $(-1)^8$ by summing the exponents.
Step 5.1.1.1:
Multiply $-1$ by $(-1)^8$.
Step 5.1.1.1.1:
Raise $-1$ to the first power: $S = - \frac{1 + (-1)^1 \cdot (-1)^8}{1 - (-1)}$.
Step 5.1.1.1.2:
Apply the exponent rule $a^m \cdot a^n = a^{m+n}$: $S = - \frac{1 + (-1)^{1+8}}{1 - (-1)}$.
Step 5.1.1.2:
Sum the exponents: $S = - \frac{1 + (-1)^9}{1 - (-1)}$.
Step 5.1.2:
Elevate $-1$ to the ninth power: $S = - \frac{1 - 1}{1 - (-1)}$.
Step 5.1.3:
Subtract one from one: $S = - \frac{0}{1 - (-1)}$.
Step 5.2:
Streamline the denominator.
Step 5.2.1:
Multiply $-1$ by $-1$: $S = - \frac{0}{1 + 1}$.
Step 5.2.2:
Add one to one: $S = - \frac{0}{2}$.
Step 5.3:
Divide zero by two: $S = -0$.
Step 5.4:
Multiply $-1$ by zero: $S = 0$.
Knowledge Notes:
Geometric Series: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). The sum of the first $n$ terms of a geometric series can be calculated using the formula $S = a \left( \frac{1 - r^n}{1 - r} \right)$, where $a$ is the first term and $n$ is the number of terms.
Common Ratio: The common ratio in a geometric series is the constant factor between consecutive terms. It is calculated by dividing any term by the preceding term: $r = \frac{a_{n+1}}{a_n}$.
Exponent Rules: When multiplying powers with the same base, you can add the exponents: $a^m \cdot a^n = a^{m+n}$. When raising a power to a power, you multiply the exponents: $(a^m)^n = a^{m \cdot n}$.
Negative Exponents: A negative exponent indicates that the base is on the wrong side of a fraction and needs to be inverted. For example, $a^{-n} = \frac{1}{a^n}$.
Simplification: The process of simplifying an expression involves combining like terms, reducing fractions, and performing arithmetic operations to achieve the simplest form of the expression.
Finite Geometric Series: The sum of a finite geometric series is found using the formula provided, and it only works when the absolute value of the common ratio is less than one ($|r| < 1$). In this problem, the series is finite and has a common ratio of $-1$.
Even and Odd Powers: Raising $-1$ to an even power results in $1$, while raising it to an odd power results in $-1$. This is due to the fact that an even number of negative factors will always multiply to a positive result, and an odd number will result in a negative.
