Evaluate the Summation sum from k=0 to 3 of (-1/2)^k
The question asks for the evaluation of a finite mathematical series. Specifically, it requires you to compute the sum of a sequence of values that follow the formula (-1/2)^k, where k is an index that increases from 0 to 3. The term (-1/2)^k represents each term in the series, and as k increases, the exponent changes, which affects the value of each term. The sum of these values from k=0 to k=3 is to be calculated.
$\sum_{k = 0}^{3} \left(\left(\right. - \frac{1}{2} \left.\right)\right)^{k}$
Answer
Solution:
Step:1
Write out the summation for each term where $k$ ranges from $0$ to $3$.
$(-\frac{1}{2})^{0} + (-\frac{1}{2})^{1} + (-\frac{1}{2})^{2} + (-\frac{1}{2})^{3}$
Step:2
Begin simplification process.
Step:2.1
Break down each term individually.
Step:2.1.1
Utilize the exponentiation rule $(ab)^{n} = a^{n}b^{n}$ to separate the terms.
$(-1)^{0}(\frac{1}{2})^{0} + (-\frac{1}{2})^{1} + (-\frac{1}{2})^{2} + (-\frac{1}{2})^{3}$
Step:2.1.2
Recognize that any number to the zero power equals one.
$1(\frac{1}{2})^{0} + (-\frac{1}{2})^{1} + (-\frac{1}{2})^{2} + (-\frac{1}{2})^{3}$
Step:2.1.3
Calculate the first term.
$\frac{1}{2^{0}} + (-\frac{1}{2})^{1} + (-\frac{1}{2})^{2} + (-\frac{1}{2})^{3}$
Step:2.1.4
Simplify the first term to one.
$1 + (-\frac{1}{2})^{1} + (-\frac{1}{2})^{2} + (-\frac{1}{2})^{3}$
Step:2.1.5
Evaluate the second term.
$1 - \frac{1}{2} + (-\frac{1}{2})^{2} + (-\frac{1}{2})^{3}$
Step:2.1.6
Apply the exponentiation rule again for the third term.
$1 - \frac{1}{2} + (-1)^{2}(\frac{1}{2})^{2} + (-\frac{1}{2})^{3}$
Step:2.1.7
Simplify the third term.
$1 - \frac{1}{2} + \frac{1}{4} + (-\frac{1}{2})^{3}$
Step:2.1.8
Apply the exponentiation rule for the fourth term.
$1 - \frac{1}{2} + \frac{1}{4} + (-1)^{3}(\frac{1}{2})^{3}$
Step:2.1.9
Simplify the fourth term.
$1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8}$
Step:2.2
Combine the terms over a common denominator.
Step:2.2.1
Express the number one as a fraction.
$\frac{1}{1} - \frac{1}{2} + \frac{1}{4} - \frac{1}{8}$
Step:2.2.2
Find a common denominator for all terms.
$\frac{8}{8} - \frac{4}{8} + \frac{2}{8} - \frac{1}{8}$
Step:2.3
Add and subtract the numerators over the common denominator.
$\frac{8 - 4 + 2 - 1}{8}$
Step:2.4
Simplify the numerator.
$\frac{5}{8}$
Step:3
Present the result in various forms.
Exact Form: $\frac{5}{8}$ Decimal Form: $0.625$
Knowledge Notes:
Summation Notation: The summation notation $\sum$ is used to denote the sum of terms in a sequence. The expression $\sum_{k=a}^{b} f(k)$ means that we should evaluate the function $f(k)$ for every integer $k$ from $a$ to $b$ and add the results together.
Exponentiation Rules: The power rule $(ab)^{n} = a^{n}b^{n}$ allows us to distribute the exponent over a product. Another rule is that any number raised to the power of zero is equal to one.
Negative Exponents: Negative bases raised to an even exponent result in a positive value, while negative bases raised to an odd exponent result in a negative value.
Common Denominator: When adding or subtracting fractions, it is necessary to have a common denominator. This allows us to combine the numerators while keeping the denominator the same.
Simplifying Fractions: Once we have a common denominator, we can add or subtract the numerators as if they were whole numbers, then place the result over the common denominator to get the simplified fraction.
Decimal Representation: Fractions can be converted to decimal form by performing the division of the numerator by the denominator. This can be useful for understanding the size of the number in a more familiar format.
