Evaluate the Summation sum from j=1 to 4 of (1/4)^j
The problem provided is asking to calculate the sum of a series where the variable j takes on values from 1 to 4. For each value of j, you are to calculate the expression (1/4)^j and then add all of these computed values together to get the total sum of the series. This is a geometric series where each term is a power of 1/4. The problem does not request the individual terms to be listed, only the final summed value of all terms in the series.
$\sum_{j = 1}^{4} \left(\left(\right. \frac{1}{4} \left.\right)\right)^{j}$
Answer
Solution:
Step 1: Write out the series for each value of $j$.
$$\left(\frac{1}{4}\right)^1 + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \left(\frac{1}{4}\right)^4$$
Step 2: Simplify the expression.
Step 2.1: Simplify each term individually.
- $\left(\frac{1}{4}\right)^1 = \frac{1}{4}$
- $\left(\frac{1}{4}\right)^2 = \frac{1}{4^2}$
- $\left(\frac{1}{4}\right)^3 = \frac{1}{4^3}$
- $\left(\frac{1}{4}\right)^4 = \frac{1}{4^4}$
Step 2.2: Find a common denominator for all terms.
Multiply $\frac{1}{4}$ by $\frac{64}{64}$ to get $\frac{64}{256}$.
Multiply $\frac{1}{4^2}$ by $\frac{16}{16}$ to get $\frac{16}{256}$.
Multiply $\frac{1}{4^3}$ by $\frac{4}{4}$ to get $\frac{4}{256}$.
The term $\frac{1}{4^4}$ is already $\frac{1}{256}$.
Step 2.3: Combine the numerators over the common denominator.
$$\frac{64 + 16 + 4 + 1}{256}$$
Step 2.4: Perform the addition.
$$\frac{85}{256}$$
Step 3: Present the result in various forms.
- Exact Form: $\frac{85}{256}$
- Decimal Form: $0.33203125$
Knowledge Notes:
The problem involves evaluating a finite geometric series. 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. In this case, the common ratio is $\frac{1}{4}$. The sum of a finite geometric series can be found by adding the individual terms, which is what we did in this problem.
The steps involved in solving the problem include:
Expanding the series to show each term.
Simplifying each term by raising the common ratio to the respective power of $j$.
Finding a common denominator to combine the terms, which is necessary when adding fractions with different denominators.
Adding the numerators together while keeping the common denominator.
Presenting the result in both exact (fractional) form and decimal form.
This problem also requires knowledge of exponentiation rules, such as any number raised to the power of 1 is itself, and 1 raised to any power is still 1. Additionally, understanding how to manipulate fractions, such as finding a common denominator and simplifying fractions, is essential.
