Evaluate the Summation sum from k=1 to 9 of 2^(k-1)

Solution:

Step 1:

To calculate the sum of a finite geometric series, we use the formula $S = a \left(\frac{1 - r^{n}}{1 - r}\right)$, where $S$ is the sum, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Step 2:

Determine the common ratio $r$ by using the definition $r = \frac{a_{k+1}}{a_k}$.

Step 2.1:

Insert $a_{k}$ and $a_{k+1}$ into the ratio formula: $r = \frac{2^{(k+1)-1}}{2^{k-1}}$.

Step 2.2:

Proceed to simplify the expression.

Step 2.2.1:

Eliminate the shared base of $2^{k-1}$ from both numerator and denominator.

Step 2.2.1.1:

Extract $2^{k-1}$ from the numerator: $r = \frac{2^{k-1} \cdot 2^{(k+1)-(k-1)}}{2^{k-1}}$.

Step 2.2.1.2:

Cancel out identical factors.

Step 2.2.1.2.1:

Introduce multiplication by $1$: $r = \frac{2^{k-1} \cdot 2^{(k+1)-(k-1)}}{2^{k-1} \cdot 1}$.

Step 2.2.1.2.2:

Remove the common $2^{k-1}$ factor: $r = \frac{\cancel{2^{k-1}} \cdot 2^{(k+1)-(k-1)}}{\cancel{2^{k-1}} \cdot 1}$.

Step 2.2.1.2.3:

Rewrite the simplified expression: $r = \frac{2^{(k+1)-(k-1)}}{1}$.

Step 2.2.1.2.4:

Divide by $1$: $r = 2^{(k+1)-(k-1)}$.

Step 2.2.2:

Combine like terms: $r = 2^{(k-k+1)}$.

Step 2.2.3:

Simplify the exponent.

Step 2.2.3.1:

Apply the distributive property: $r = 2^{1}$.

Step 2.2.4:

Subtract $k$ from itself: $r = 2^{0+1}$.

Step 2.2.5:

Combine the terms: $r = 2^{1}$.

Step 2.2.6:

Calculate the power of $2$: $r = 2$.

Step 3:

Identify the first term $a$ by substituting $k=1$ into $2^{k-1}$.

Step 3.1:

Replace $k$ with $1$: $a = 2^{1-1}$.

Step 3.2:

Simplify the term.

Step 3.2.1:

Subtract within the exponent: $a = 2^{0}$.

Step 3.2.2:

Recognize that any number to the power of zero is one: $a = 1$.

Step 4:

Insert the values for $r$, $a$, and $n$ into the sum formula: $S = 1 \left(\frac{1 - 2^{9}}{1 - 2}\right)$.

Step 5:

Simplify the expression.

Step 5.1:

Multiply the fraction by $1$: $\frac{1 - 2^{9}}{1 - 2}$.

Step 5.2:

Work on the numerator.

Step 5.2.1:

Compute $2$ raised to the ninth power: $\frac{1 - 512}{1 - 2}$.

Step 5.2.2:

Apply multiplication: $\frac{1 - 512}{1 - 2}$.

Step 5.2.3:

Subtract within the numerator: $\frac{-511}{1 - 2}$.

Step 5.3:

Simplify the denominator.

Step 5.3.1:

Multiply within the denominator: $\frac{-511}{-1}$.

Step 5.3.2:

Subtract in the denominator: $\frac{-511}{-1}$.

Step 5.4:

Divide the numerator by the denominator: $511$.

Knowledge Notes:

1. Geometric Series: A geometric series is a series with a constant ratio between successive terms. The sum of a finite geometric series can be found using the formula $S = a \left(\frac{1 - r^{n}}{1 - r}\right)$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

2. Exponents: When dividing terms with the same base, subtract the exponents. For example, $\frac{2^{m}}{2^{n}} = 2^{m-n}$. Any number raised to the power of zero is one: $2^{0} = 1$.

3. Simplification: Simplifying expressions involves combining like terms, canceling common factors, and performing arithmetic operations to reduce the expression to its simplest form.

4. Summation of Series: When evaluating the summation of a series, it's important to identify the pattern of the series (arithmetic, geometric, etc.) and apply the appropriate formula to find the sum.

5. Negative Exponents: A negative exponent represents the reciprocal of the base raised to the positive exponent. For example, $2^{-n} = \frac{1}{2^{n}}$.

6. Distributive Property: The distributive property allows you to multiply a sum by multiplying each addend separately and then add the products. It is expressed as $a(b + c) = ab + ac$.