Evaluate the Summation sum from j=1 to 9 of (-1/4)^j

Solution:

Step 1: Understanding the Geometric Series

To calculate the sum of a finite geometric series, we use the formula $S = a \left(\frac{1 - r^n}{1 - r}\right)$, where:

• $S$ is the sum of the series,
• $a$ is the first term of the series,
• $r$ is the common ratio (the factor by which we multiply each term to get the next term),
• $n$ is the number of terms in the series.

Step 2: Determining the Common Ratio

To find the common ratio $r$, we use the relationship $r = \frac{a_{j+1}}{a_j}$.

Step 2.1: Applying the Ratio Formula

Insert the terms $a_j = \left(-\frac{1}{4}\right)^j$ and $a_{j+1} = \left(-\frac{1}{4}\right)^{j+1}$ into the ratio formula:

$r = \frac{\left(-\frac{1}{4}\right)^{j+1}}{\left(-\frac{1}{4}\right)^j}$.

Step 2.2: Simplifying the Ratio

Step 2.2.1: Factoring Out Common Terms

Extract $\left(-\frac{1}{4}\right)^j$ from the numerator:

$r = \frac{\left(-\frac{1}{4}\right)^j \cdot \left(-\frac{1}{4}\right)}{\left(-\frac{1}{4}\right)^j}$.

Step 2.2.2: Canceling Out Common Factors

Step 2.2.2.1: Multiplying by 1

Express the denominator as a product with 1:

$r = \frac{\left(-\frac{1}{4}\right)^j \cdot \left(-\frac{1}{4}\right)}{\left(-\frac{1}{4}\right)^j \cdot 1}$.

Step 2.2.2.2: Performing the Cancellation

Eliminate the common factor:

$r = \frac{\cancel{\left(-\frac{1}{4}\right)^j} \cdot \left(-\frac{1}{4}\right)}{\cancel{\left(-\frac{1}{4}\right)^j} \cdot 1}$.

Step 2.2.2.3: Finalizing the Ratio

Simplify the expression:

$r = \frac{-\frac{1}{4}}{1}$.

Step 2.2.2.4: Dividing by 1

Conclude that $r = -\frac{1}{4}$.

Step 3: Identifying the First Term

To find the first term $a$, substitute $j = 1$ into the term formula:

$a = \left(-\frac{1}{4}\right)^1$.

Step 3.1: Substitution

Replace $j$ with 1:

$a = -\frac{1}{4}$.

Step 4: Plugging Values into the Sum Formula

Insert $a$, $r$, and $n = 9$ into the sum formula:

$S = -\frac{1}{4} \cdot \frac{1 - \left(-\frac{1}{4}\right)^9}{1 - (-\frac{1}{4})}$.

Step 5: Simplifying the Sum

Step 5.1: Working on the Numerator

Step 5.1.1: Applying the Power Rule

Utilize the power rule $(ab)^n = a^n b^n$:

$S = -\frac{1}{4} \cdot \frac{1 - ((-1)^9 (\frac{1}{4})^9)}{1 - (-\frac{1}{4})}$.

Step 5.1.2: Multiplying by $-1$

Combine the negative signs:

$S = -\frac{1}{4} \cdot \frac{1 + (-1)^{10} \frac{1}{4^9}}{1 + \frac{1}{4}}$.

Step 5.2: Simplifying the Denominator

Combine the terms over a common denominator:

$S = -\frac{1}{4} \cdot \frac{\frac{262145}{262144}}{\frac{5}{4}}$.

Step 5.3: Multiplying by the Reciprocal

Multiply the numerator by the reciprocal of the denominator:

$S = -\frac{1}{4} \left(\frac{262145}{262144} \cdot \frac{4}{5}\right)$.

Step 5.4: Canceling Common Factors

Eliminate the common factor of 5:

$S = -\frac{1}{4} \left(\frac{52429}{262144} \cdot 4\right)$.

Step 5.5: Final Simplification

Cancel the common factor of 4:

$S = -\frac{52429}{262144}$.

Step 6: Presenting the Result

The sum can be expressed in different forms:

Exact Form: $- \frac{52429}{262144}$ Decimal Form: $- 0.20000076 \ldots$

Knowledge Notes:

• 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.

• The sum of the first $n$ terms of a geometric series is given by 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.

• Simplifying expressions often involves factoring out common terms, canceling them, and applying power rules such as $(ab)^n = a^n b^n$ and $a^{m+n} = a^m a^n$.

• When dealing with negative exponents or bases, it's important to carefully consider the signs and apply the rules for exponents correctly.

• The final result of a series can be expressed in exact form (as a fraction) or in decimal form. The exact form is more precise and should be used when exact values are necessary.