Evaluate the Summation sum from i=1 to 6 of 1/6*3^i

Solution:

Step:1

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

Step:2

Determine the common ratio ($r$) by using the formula $r = \frac{a_{i + 1}}{a_{i}}$.

Step:2.1

Insert the values for $a_{i}$ and $a_{i + 1}$ into the ratio formula: $r = \frac{\frac{1}{6} \cdot 3^{i + 1}}{\frac{1}{6} \cdot 3^{i}}$.

Step:2.2

Proceed to simplify the expression.

Step:2.2.1

Eliminate the identical factor of $\frac{1}{6}$.

Step:2.2.1.1

Remove the shared factor: $r = \frac{\cancel{\frac{1}{6}} \cdot 3^{i + 1}}{\cancel{\frac{1}{6}} \cdot 3^{i}}$.

Step:2.2.1.2

Reformulate the expression: $r = \frac{3^{i + 1}}{3^{i}}$.

Step:2.2.2

Remove the common powers of $3^{i}$ and $3^{i + 1}$.

Step:2.2.2.1

Factor out $3^{i}$ from $3^{i + 1}$: $r = \frac{3^{i} \cdot 3}{3^{i}}$.

Step:2.2.2.2

Eliminate the identical factors.

Step:2.2.2.2.1

Introduce a factor of $1$: $r = \frac{3^{i} \cdot 3}{3^{i} \cdot 1}$.

Step:2.2.2.2.2

Remove the shared factor: $r = \frac{\cancel{3^{i}} \cdot 3}{\cancel{3^{i}} \cdot 1}$.

Step:2.2.2.2.3

Rephrase the expression: $r = \frac{3}{1}$.

Step:2.2.2.2.4

Divide $3$ by $1$: $r = 3$.

Step:3

Identify the first term ($a$) by substituting the initial index value and simplifying.

Step:3.1

Replace $i$ with $1$ in $\frac{1}{6} \cdot 3^{i}$: $a = \frac{1}{6} \cdot 3^{1}$.

Step:3.2

Simplify the term.

Step:3.2.1

Compute the power of $3$: $a = \frac{1}{6} \cdot 3$.

Step:3.2.2

Eliminate the common factor of $3$.

Step:3.2.2.1

Extract $3$ from $6$: $a = \frac{1}{3(2)} \cdot 3$.

Step:3.2.2.2

Cancel out the shared factor: $a = \frac{1}{\cancel{3} \cdot 2} \cdot \cancel{3}$.

Step:3.2.2.3

Reformulate the expression: $a = \frac{1}{2}$.

Step:4

Insert the values for the common ratio, the first term, and the number of terms into the summation formula: $\frac{1}{2} \cdot \frac{1 - 3^{6}}{1 - 3}$.

Step:5

Carry out the simplification process.

Step:5.1

Simplify the numerator.

Step:5.1.1

Exponentiate $3$ to the sixth power: $\frac{1}{2} \cdot \frac{1 - 729}{1 - 3}$.

Step:5.1.2

Multiply $-1$ by $729$: $\frac{1}{2} \cdot \frac{1 - 729}{1 - 3}$.

Step:5.1.3

Subtract $729$ from $1$: $\frac{1}{2} \cdot \frac{-728}{1 - 3}$.

Step:5.2

Simplify the denominator.

Step:5.2.1

Multiply $-1$ by $3$: $\frac{1}{2} \cdot \frac{-728}{1 - 3}$.

Step:5.2.2

Subtract $3$ from $1$: $\frac{1}{2} \cdot \frac{-728}{-2}$.

Step:5.3

Remove the common factor of $2$.

Step:5.3.1

Factor $2$ out of $-728$: $\frac{1}{2} \cdot \frac{2(-364)}{-2}$.

Step:5.3.2

Cancel out the shared factor: $\frac{1}{\cancel{2}} \cdot \frac{\cancel{2} \cdot -364}{-2}$.

Step:5.3.3

Rephrase the expression: $\frac{-364}{-2}$.

Step:5.4

Divide $-364$ by $-2$: $182$.

Knowledge Notes:

To solve this problem, we need to understand the concept of a geometric series and how to calculate its sum. A geometric series 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 sum of the first $n$ terms of a geometric series can be calculated 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.

In this problem, we are given a geometric series with a common ratio of $3$ and a first term of $\frac{1}{2}$. The series has $6$ terms. We use the formula for the sum of a geometric series to find the total sum. Simplifying the expression involves basic algebraic steps such as canceling out common factors and performing arithmetic operations like exponentiation, multiplication, and division. The final result is the sum of the series, which in this case is $182$.