Evaluate the Summation sum from i=1 to 4 of (1/3)^(i-1)

Solution:

Step:1

Write out the terms of the summation for $i$ ranging from $1$ to $4$.

$$(\frac{1}{3}{)}^{1-1}+(\frac{1}{3}{)}^{2-1}+(\frac{1}{3}{)}^{3-1}+(\frac{1}{3}{)}^{4-1}$$

Step:2

Proceed to simplify the expression.

Step:2.1

Evaluate each term individually.

Step:2.1.1

Calculate $1-1$.

$$(\frac{1}{3}{)}^0+(\frac{1}{3}{)}^{2-1}+(\frac{1}{3}{)}^{3-1}+(\frac{1}{3}{)}^{4-1}$$

Step:2.1.2

Apply the exponent rule to $\frac{1}{3}$.

$$\frac{1^0}{3^0}+(\frac{1}{3}{)}^{2-1}+(\frac{1}{3}{)}^{3-1}+(\frac{1}{3}{)}^{4-1}$$

Step:2.1.3

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

$$\frac{1}{3^0}+(\frac{1}{3}{)}^{2-1}+(\frac{1}{3}{)}^{3-1}+(\frac{1}{3}{)}^{4-1}$$

Step:2.1.4

Simplify the first term to $1$.

$$1+(\frac{1}{3}{)}^{2-1}+(\frac{1}{3}{)}^{3-1}+(\frac{1}{3}{)}^{4-1}$$

Step:2.1.5

Evaluate $2-1$.

$$1+(\frac{1}{3}{)}^1+(\frac{1}{3}{)}^{3-1}+(\frac{1}{3}{)}^{4-1}$$

Step:2.1.6

Simplify the second term.

$$1+\frac{1}{3}+(\frac{1}{3}{)}^{3-1}+(\frac{1}{3}{)}^{4-1}$$

Step:2.1.7

Calculate $3-1$.

$$1+\frac{1}{3}+(\frac{1}{3}{)}^2+(\frac{1}{3}{)}^{4-1}$$

Step:2.1.8

Apply the exponent rule to the third term.

$$1+\frac{1}{3}+\frac{1^2}{3^2}+(\frac{1}{3}{)}^{4-1}$$

Step:2.1.9

Simplify the third term.

$$1+\frac{1}{3}+\frac{1}{3^2}+(\frac{1}{3}{)}^{4-1}$$

Step:2.1.10

Evaluate $4-1$.

$$1+\frac{1}{3}+\frac{1}{9}+(\frac{1}{3}{)}^3$$

Step:2.1.11

Apply the exponent rule to the fourth term.

$$1+\frac{1}{3}+\frac{1}{9}+\frac{1^3}{3^3}$$

Step:2.1.12

Simplify the fourth term.

$$1+\frac{1}{3}+\frac{1}{9}+\frac{1}{3^3}$$

Step:2.1.13

Combine the simplified terms.

$$1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}$$

Step:2.2

Find a common denominator for the fractions.

Step:2.2.1

Express $1$ as a fraction with denominator $27$.

$$\frac{27}{27}+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}$$

Step:2.2.2

Convert $\frac{1}{3}$ to have the common denominator $27$.

$$\frac{27}{27}+\frac{9}{27}+\frac{1}{9}+\frac{1}{27}$$

Step:2.2.3

Convert $\frac{1}{9}$ to have the common denominator $27$.

$$\frac{27}{27}+\frac{9}{27}+\frac{3}{27}+\frac{1}{27}$$

Step:2.2.4

Combine the fractions over the common denominator.

$$\frac{27+9+3+1}{27}$$

Step:2.3

Add the numerators.

$$\frac{40}{27}$$

Step:3

Present the final result in various forms.

Exact Form: $\frac{40}{27}$ Decimal Form: $1.4815...$ Mixed Number Form: $1\frac{13}{27}$

Knowledge Notes:

This 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 $\frac{1}{3}$.

The steps of the solution involve:

1. Expanding the series based on the given summation limits.

2. Simplifying each term by applying exponent rules, such as any number to the power of zero equals one.

3. Finding a common denominator to combine the terms, which is necessary when adding fractions with different denominators.

4. Adding the numerators over the common denominator to find the sum of the series.

5. Presenting the result in various forms, including exact fraction, decimal, and mixed number forms.

This problem also illustrates the importance of understanding the properties of exponents and the process of adding fractions, which are fundamental concepts in algebra.