Evaluate the Summation sum from k=0 to 2 of 12(-1/2)^k
The problem is asking to calculate the sum of a finite geometric series. The series is defined by summing terms that follow a specific pattern, each of which is the previous term multiplied by a common ratio. In this case, the first term of the series is 12, the common ratio is -1/2, and the series is to be summed from k=0 to k=2. The task is to find the total value after summing the three terms of this series according to the given formula.
$\sum_{k = 0}^{2} 12 \left(\left(\right. - \frac{1}{2} \left.\right)\right)^{k}$
Answer
Solution:
Step 1: Expand the Summation
Write out each term of the summation for $k = 0, 1, 2$.
$$12\left(-\frac{1}{2}\right)^0 + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2: Simplify the Expression
Step 2.1: Simplify Each Term Individually
Step 2.1.1: Apply the Power Rule
Use the power rule $(ab)^n = a^n b^n$ to separate the exponent over the product.
Step 2.1.1.1: Simplify the First Term
Break down the first term using the power rule.
$$12\left((-1)^0\left(\frac{1}{2}\right)^0\right) + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.1.2: Simplify the Remaining Terms
Apply the same rule to the other terms.
$$12\left((-1)^0\frac{1^0}{2^0}\right) + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.2: Recognize that Any Number Raised to 0 is 1
Replace the expressions raised to the power of 0 with 1.
$$12\left(1\frac{1^0}{2^0}\right) + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.3: Simplify the Fraction
Multiply by the simplified fraction.
$$12\frac{1^0}{2^0} + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.4: Replace 1 Raised to Any Power with 1
Replace the numerators raised to the power of 0 with 1.
$$12\frac{1}{2^0} + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.5: Recognize that 2 Raised to the Power of 0 is 1
Replace $2^0$ with 1.
$$12\left(\frac{1}{1}\right) + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.6: Simplify the Expression
Step 2.1.6.1: Cancel Common Factors
Remove the common factors of 1.
$$12\left(\frac{\cancel{1}}{\cancel{1}}\right) + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.6.2: Rewrite the Expression
Express the simplified terms.
$$12 \cdot 1 + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.7: Multiply 12 by 1
Multiply the first term.
$$12 + 12\left(-\frac{1}{2}\right)^1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.8: Simplify the Negative Fraction
Simplify the second term.
$$12 + 12\left(-\frac{1}{2}\right) + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.9: Cancel the Common Factor of 2
Step 2.1.9.1: Move the Negative to the Numerator
Rewrite the negative fraction.
$$12 + 12\left(\frac{-1}{2}\right) + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.9.2: Factor 2 out of 12
Factor out the 2 from the second term.
$$12 + 2(6)\frac{-1}{2} + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.9.3: Cancel the Common Factor
Remove the common factor of 2.
$$12 + \cancel{2} \cdot 6\frac{-1}{\cancel{2}} + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.9.4: Rewrite the Expression
Express the simplified terms.
$$12 + 6 \cdot -1 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.10: Multiply 6 by -1
Perform the multiplication.
$$12 - 6 + 12\left(-\frac{1}{2}\right)^2$$
Step 2.1.11: Apply the Power Rule Again
Use the power rule to distribute the exponent in the third term.
Step 2.1.11.1: Apply the Product Rule to $-\frac{1}{2}$
Break down the third term using the power rule.
$$12 - 6 + 12\left((-1)^2\left(\frac{1}{2}\right)^2\right)$$
Step 2.1.11.2: Apply the Product Rule to $\frac{1}{2}$
Apply the same rule to the fraction.
$$12 - 6 + 12\left((-1)^2\frac{1^2}{2^2}\right)$$
Step 2.1.12: Raise -1 to the Power of 2
Square the -1.
$$12 - 6 + 12\left(1\frac{1^2}{2^2}\right)$$
Step 2.1.13: Multiply the Fraction by 1
Simplify the multiplication.
$$12 - 6 + 12\frac{1^2}{2^2}$$
Step 2.1.14: Simplify the Numerator
Replace $1^2$ with 1.
$$12 - 6 + 12\frac{1}{2^2}$$
Step 2.1.15: Simplify the Denominator
Square the 2.
$$12 - 6 + 12\left(\frac{1}{4}\right)$$
Step 2.1.16: Cancel the Common Factor of 4
Step 2.1.16.1: Factor 4 out of 12
Factor out the 4 from the third term.
$$12 - 6 + 4(3)\frac{1}{4}$$
Step 2.1.16.2: Cancel the Common Factor
Remove the common factor of 4.
$$12 - 6 + \cancel{4} \cdot 3\frac{1}{\cancel{4}}$$
Step 2.1.16.3: Rewrite the Expression
Express the simplified terms.
$$12 - 6 + 3$$
Step 2.2: Subtract 6 from 12
Combine the first two terms.
$$6 + 3$$
Step 2.3: Add 6 and 3
Sum the final terms.
$$9$$
Knowledge Notes:
To solve this problem, we used several mathematical concepts:
Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of terms. The sequence is defined by an index that varies from a lower bound to an upper bound.
Exponent Rules: The exponent rules, such as $(ab)^n = a^n b^n$ and $a^0 = 1$, are fundamental in simplifying expressions involving powers.
Negative Numbers: When dealing with negative numbers, especially when they are raised to powers, it's important to remember that a negative number raised to an even power results in a positive number, while a negative number raised to an odd power results in a negative number.
Fraction Simplification: Simplifying fractions involves factoring out common factors from the numerator and the denominator or multiplying/dividing both by the same number.
Basic Arithmetic: The problem also involves basic arithmetic operations such as addition, subtraction, and multiplication.
By applying these concepts step by step, we can simplify the given summation and find the result.
