Evaluate the Summation sum from k=1 to 3 of (1/2)^(k+1)
The question asks to perform a calculation of a finite series by evaluating the summation of a given formula, where the summation index k starts at 1 and ends at 3. The formula provided for the summation is (1/2)^(k+1), which means for each value of k within the specified range, calculate (1/2) raised to the power of k+1, and then sum these values to get the final result.
$\sum_{k = 1}^{3} \left(\left(\right. \frac{1}{2} \left.\right)\right)^{k + 1}$
Answer
Solution:
Step 1: Write out the summation terms
Calculate the value of the series by substituting $k = 1, 2, 3$ into the expression $\left(\frac{1}{2}\right)^{k+1}$.
$$\left(\frac{1}{2}\right)^{1+1} + \left(\frac{1}{2}\right)^{2+1} + \left(\frac{1}{2}\right)^{3+1}$$
Step 2: Simplify the series
Step 2.1: Break down each term
Step 2.1.1: Combine $1$ and $1$
$$\left(\frac{1}{2}\right)^{2} + \left(\frac{1}{2}\right)^{3} + \left(\frac{1}{2}\right)^{4}$$
Step 2.1.2: Apply exponent rules to $\frac{1}{2}$
$$\frac{1}{2^2} + \left(\frac{1}{2}\right)^{3} + \left(\frac{1}{2}\right)^{4}$$
Step 2.1.3: Recognize that $1$ to any power is $1$
$$\frac{1}{4} + \left(\frac{1}{2}\right)^{3} + \left(\frac{1}{2}\right)^{4}$$
Step 2.1.4: Calculate $2$ to the power of $2$
$$\frac{1}{4} + \frac{1}{2^3} + \left(\frac{1}{2}\right)^{4}$$
Step 2.1.5: Combine $2$ and $1$
$$\frac{1}{4} + \frac{1}{8} + \left(\frac{1}{2}\right)^{4}$$
Step 2.1.6: Apply exponent rules to $\frac{1}{2}$
$$\frac{1}{4} + \frac{1}{8} + \frac{1}{2^4}$$
Step 2.1.7: Recognize that $1$ to any power is $1$
$$\frac{1}{4} + \frac{1}{8} + \frac{1}{16}$$
Step 2.2: Find a common denominator
Step 2.2.1: Multiply $\frac{1}{4}$ by $\frac{4}{4}$
$$\frac{4}{16} + \frac{1}{8} + \frac{1}{16}$$
Step 2.2.2: Multiply $\frac{1}{8}$ by $\frac{2}{2}$
$$\frac{4}{16} + \frac{2}{16} + \frac{1}{16}$$
Step 2.3: Combine the numerators over the common denominator
$$\frac{4 + 2 + 1}{16}$$
Step 2.4: Add the numerators
$$\frac{7}{16}$$
Step 3: Present the result in various forms
Exact Form: $\frac{7}{16}$ Decimal Form: $0.4375$
Knowledge Notes:
The problem involves evaluating a finite geometric series, which 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. In this case, the common ratio is $\left(\frac{1}{2}\right)$. The steps of the solution involve:
Expansion: Writing out each term of the series explicitly.
Simplification: Breaking down each term by applying exponent rules and simplifying.
Common Denominator: Finding a common denominator to combine the terms, which is necessary when adding fractions.
Combining Terms: Adding the numerators once a common denominator is found.
Final Result: Presenting the result in both exact (fractional) and decimal forms.
Key concepts used in this problem include exponent rules, such as $a^{m+n} = a^m \cdot a^n$ and $a^{-n} = \frac{1}{a^n}$, and the understanding of how to manipulate and simplify fractions. The product rule of exponents states that when multiplying two powers that have the same base, you can add the exponents. Additionally, the problem requires knowledge of how to find a common denominator and combine fractions, which is a fundamental concept in arithmetic.
