Evaluate the Summation sum from i=1 to 5 of 4/3*4^i

Solution:

Step 1: Write out the summation term by term

The summation given is $\sum _{i=1}^5\frac{4}{3}\cdot 4^i$. We write out each term as follows:

$\frac{4}{3}\cdot 4^1+\frac{4}{3}\cdot 4^2+\frac{4}{3}\cdot 4^3+\frac{4}{3}\cdot 4^4+\frac{4}{3}\cdot 4^5$

Step 2: Calculate the sum of the series

Upon calculating the sum of the terms, we find that the total is $\frac{5456}{3}$.

Step 3: Present the sum in various forms

The sum can be expressed in different ways:

• Exact Form: $\frac{5456}{3}$
• Decimal Form: $1818.666\dots$
• Mixed Number Form: $1818\frac{2}{3}$

Knowledge Notes:

When evaluating a summation, especially one that involves a geometric series or a pattern, the following knowledge points are relevant:

1. Summation Notation: Understanding the sigma notation $\sum$ which indicates a sum over a range of values. In this case, $\sum _{i=1}^5$ means we sum the expression for $i$ starting at 1 and ending at 5.

2. Geometric Series: The given problem involves a geometric series where each term is a constant multiple of the previous term. The general form of a geometric series is $\sum _{n=0}^{\mathrm{\infty }}a{r}^n$, where $a$ is the first term and $r$ is the common ratio.

3. Exponentiation: Recognizing that $4^i$ represents 4 raised to the power of $i$. Exponentiation is a repeated multiplication process.

4. Simplification: Combining like terms and simplifying expressions using arithmetic operations. In this case, we multiply the constant $\frac{4}{3}$ with each power of 4.

5. Fractional Forms: Converting between different representations of numbers, such as improper fractions, decimals, and mixed numbers.

6. Arithmetic Operations: Performing basic arithmetic operations such as addition, multiplication, and division with fractions and decimals.

In solving the problem, we applied these concepts to expand the series, calculate the sum, and express the result in various numerical forms.