Evaluate the Summation sum from k=0 to 9 of 2^k

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 $S$ 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 formula $r = \frac{a_{k+1}}{a_k}$.

Step 2.1:

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

Step 2.2:

Eliminate the common base of the exponentials.

Step 2.2.1:

Extract $2^k$ from $2^{k+1}$: $r = \frac{2^k \cdot 2}{2^k}$.

Step 2.2.2:

Remove the common terms.

Step 2.2.2.1:

Introduce a multiplication by $1$: $r = \frac{2^k \cdot 2}{2^k \cdot 1}$.

Step 2.2.2.2:

Cross out the common $2^k$: $r = \frac{\cancel{2^k} \cdot 2}{\cancel{2^k} \cdot 1}$.

Step 2.2.2.3:

Simplify the fraction: $r = \frac{2}{1}$.

Step 2.2.2.4:

Conclude that $r = 2$.

Step 3:

Identify the first term $a$ by substituting the initial value of $k$.

Step 3.1:

Plug in $k = 0$ into $2^k$: $a = 2^0$.

Step 3.2:

Recognize that any number to the power of $0$ equals $1$: $a = 1$.

Step 4:

Insert the values for $a$, $r$, and $n$ into the sum formula: $S = 1 \left(\frac{1 - 2^{10}}{1 - 2}\right)$.

Step 5:

Proceed with simplification.

Step 5.1:

Multiply the sum formula by $1$: $S = \frac{1 - 2^{10}}{1 - 2}$.

Step 5.2:

Work on the numerator.

Step 5.2.1:

Calculate $2^{10}$: $S = \frac{1 - 1024}{1 - 2}$.

Step 5.2.2:

Apply the negative sign: $S = \frac{1 - 1024}{1 - 2}$.

Step 5.2.3:

Complete the subtraction: $S = \frac{-1023}{1 - 2}$.

Step 5.3:

Simplify the denominator.

Step 5.3.1:

Multiply $-1$ by $2$: $S = \frac{-1023}{-1}$.

Step 5.3.2:

Finish the subtraction: $S = \frac{-1023}{-1}$.

Step 5.4:

Divide $-1023$ by $-1$ to get the sum: $S = 1023$.

Knowledge Notes:

The problem involves evaluating the sum of 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 ($r$). The formula to find the sum of the first $n$ terms of a geometric series is $S = a \left(\frac{1 - r^{n}}{1 - r}\right)$, where:

• $S$ is the sum of the series,
• $a$ is the first term of the series,
• $r$ is the common ratio, and
• $n$ is the number of terms.

In this problem, the series is $2^0 + 2^1 + 2^2 + \ldots + 2^9$, which is a geometric series with a common ratio of $2$ and the first term $a = 2^0 = 1$. The number of terms $n$ is $10$ because we start counting from $k=0$ to $k=9$, which includes $10$ terms.

The steps involve finding the common ratio $r$, identifying the first term $a$, and then substituting these values into the sum formula. After substitution, simplification is done by performing exponentiation, multiplication, and subtraction as required. The final result gives the sum of the series.