Evaluate the Summation sum from j=1 to 9 of (-4)^j
The given question asks for the calculation of a specific summation expression. You are required to determine the sum of the series where each term is (-4) raised to the power of the current term's position (j), with j starting at 1 and incrementing by 1 until it reaches 9. The series you would be summing could be written explicitly as follows:
-4^1 + (-4)^2 + (-4)^3 + ... + (-4)^9
This is a finite series with a pattern that alternates between positive and negative terms, since raising a negative number to an even power gives a positive result, and to an odd power gives a negative result. The requested task is to perform the sum of all terms conforming to this pattern from j equals 1 up to and including j equals 9.
$\sum_{j = 1}^{9} \left(\left(\right. - 4 \left.\right)\right)^{j}$
Answer
Solution:
Step 1: To calculate the sum of a finite geometric sequence, we 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 by applying the formula $r = \frac{a_{j+1}}{a_j}$.
Step 2.1: Insert the terms $a_j = (-4)^j$ and $a_{j+1} = (-4)^{j+1}$ into the ratio formula: $r = \frac{(-4)^{j+1}}{(-4)^j}$.
Step 2.2: Eliminate the similar factors in the numerator and denominator.
Step 2.2.1: Extract $(-4)^j$ from $(-4)^{j+1}$: $r = \frac{(-4)^j \times (-4)}{(-4)^j}$.
Step 2.2.2: Remove the common factors.
Step 2.2.2.1: Multiply by the identity element (1): $r = \frac{(-4)^j \times (-4)}{(-4)^j \times 1}$.
Step 2.2.2.2: Cross out the common factor: $r = \frac{\cancel{(-4)^j} \times (-4)}{\cancel{(-4)^j} \times 1}$.
Step 2.2.2.3: Simplify the expression: $r = \frac{-4}{1}$.
Step 2.2.2.4: Divide $-4$ by $1$: $r = -4$.
Step 3: Identify the first term by substituting $j=1$.
Step 3.1: Replace $j$ with $1$ in $(-4)^j$: $a = (-4)^1$.
Step 3.2: Calculate the value of the exponent: $a = -4$.
Step 4: Plug in the values for $a$, $r$, and $n$ into the sum formula: $S = -4 \left( \frac{1 - (-4)^9}{1 - (-4)} \right)$.
Step 5: Proceed with simplification.
Step 5.1: Work on the numerator.
Step 5.1.1: Compute $(-4)^9$: $S = -4 \left( \frac{1 - (-262144)}{1 - (-4)} \right)$.
Step 5.1.2: Apply the negative sign: $S = -4 \left( \frac{1 + 262144}{1 - (-4)} \right)$.
Step 5.1.3: Combine $1$ and $262144$: $S = -4 \left( \frac{262145}{1 - (-4)} \right)$.
Step 5.2: Simplify the denominator.
Step 5.2.1: Calculate $1 - (-4)$: $S = -4 \left( \frac{262145}{5} \right)$.
Step 5.2.2: Add $1$ and $4$: $S = -4 \left( \frac{262145}{5} \right)$.
Step 5.3: Divide $262145$ by $5$: $S = -4 \times 52429$.
Step 5.4: Multiply $-4$ by $52429$: $S = -209716$.
Knowledge Notes:
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 ratio.
Sum of Geometric Series: The sum of a finite geometric series is given by the formula $S = a \left( \frac{1 - r^n}{1 - r} \right)$, where $S$ is the sum, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
Common Ratio: In a geometric sequence, the common ratio ($r$) is the constant factor between consecutive terms, calculated by $r = \frac{a_{j+1}}{a_j}$.
Negative Bases in Exponents: When dealing with negative bases raised to an exponent, if the exponent is even, the result is positive; if the exponent is odd, the result is negative.
Simplification: The process of simplification involves reducing expressions to their simplest form by performing arithmetic operations and canceling out common factors.
