Evaluate the Summation sum from k=3 to 10 of (-4)(-2)^(k-1)

Solution:

Step:1 To calculate the sum of a finite geometric sequence, use the sum formula $S = a \left(\frac{1 - r^{n}}{1 - r}\right)$, where $a$ is the initial term and $r$ is the common ratio.

Step:2 Determine the common ratio $r$ by using the formula $r = \frac{a_{k+1}}{a_{k}}$ and carrying out the necessary calculations.

Step:2.1 Insert the terms $a_{k}$ and $a_{k+1}$ into the ratio formula: $r = \frac{-4((-2)^{k})}{-4((-2)^{k-1})}$.

Step:2.2 Proceed to simplify the expression.

Step:2.2.1 Eliminate the identical factor of $-4$: $r = \frac{\cancel{-4}((-2)^{k})}{\cancel{-4}((-2)^{k-1})}$.

Step:2.2.1.1 Remove the common factor: $r = \frac{(-2)^{k}}{(-2)^{k-1}}$.

Step:2.2.2 Cancel out the common powers of $(-2)$.

Step:2.2.2.1 Extract $(-2)^{k-1}$ from $(-2)^{k}$: $r = \frac{(-2)^{k-1} \cdot (-2)^{2 - (k - 1)}}{(-2)^{k-1}}$.

Step:2.2.2.2 Eliminate the common factors.

Step:2.2.2.2.1 Multiply by $1$: $r = \frac{(-2)^{k-1} \cdot (-2)^{2 - (k - 1)}}{(-2)^{k-1} \cdot 1}$.

Step:2.2.2.2.2 Cancel the common factor: $r = \frac{\cancel{(-2)^{k-1}} \cdot (-2)^{2 - (k - 1)}}{\cancel{(-2)^{k-1}} \cdot 1}$.

Step:2.2.2.2.3 Rewrite the expression: $r = \frac{(-2)^{2 - (k - 1)}}{1}$.

Step:2.2.2.2.4 Divide by $1$: $r = (-2)^{2 - (k - 1)}$.

Step:2.2.3 Combine $k$ and $0$: $r = (-2)^{1}$.

Step:2.2.4 Simplify the terms.

Step:2.2.4.1 Apply the distributive property: $r = (-2)^{1}$.

Step:2.2.4.2 Multiply $-1$ by $-1$: $r = (-2)^{1}$.

Step:2.2.5 Subtract $k$ from itself: $r = (-2)^{1}$.

Step:2.2.6 Add $0$ and $1$: $r = (-2)^{1}$.

Step:2.2.7 Evaluate the power of $-2$: $r = -2$.

Step:3 Calculate the first term of the series by substituting $k=3$: $a = -4((-2)^{3-1})$.

Step:3.1 Replace $k$ with $3$ in the term $-4((-2)^{k-1})$: $a = -4((-2)^{2})$.

Step:3.2 Simplify the expression.

Step:3.2.1 Subtract $1$ from $3$: $a = -4((-2)^{2})$.

Step:3.2.2 Raise $-2$ to the second power: $a = -4 \cdot 4$.

Step:3.2.3 Multiply $-4$ by $4$: $a = -16$.

Step:4 Insert the values for $r$, $a$, and the number of terms into the summation formula: $S = -16 \left(\frac{1 - (-2)^{8}}{1 - (-2)}\right)$.

Step:5 Carry out the simplification process.

Step:5.1 Work on the numerator first.

Step:5.1.1 Compute $(-2)^{8}$: $S = -16 \left(\frac{1 - 256}{1 - (-2)}\right)$.

Step:5.1.2 Multiply $-1$ by $256$: $S = -16 \left(\frac{1 - 256}{1 + 2}\right)$.

Step:5.1.3 Subtract $256$ from $1$: $S = -16 \left(\frac{-255}{3}\right)$.

Step:5.2 Simplify the denominator.

Step:5.2.1 Add $1$ and $2$: $S = -16 \left(\frac{-255}{3}\right)$.

Step:5.3 Divide $-255$ by $3$: $S = -16 \cdot -85$.

Step:5.4 Multiply $-16$ by $-85$: $S = 1360$.

Knowledge Notes:

1. Geometric Series: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of a finite geometric series can be calculated using 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.

2. Common Ratio: The common ratio in a geometric sequence is the factor by which each term is multiplied to get the next term. It is found by dividing any term by the preceding term, i.e., $r = \frac{a_{k+1}}{a_{k}}$.

3. Simplifying Expressions: When simplifying expressions, common factors can be canceled out from the numerator and denominator. Powers can be simplified using the rules of exponents, such as $a^{m} \cdot a^{n} = a^{m+n}$ and $(a^{m})^{n} = a^{m \cdot n}$.

4. Substitution: To find specific terms in a sequence, substitution involves replacing the variable with a given value and simplifying the resulting expression.

5. Arithmetic Operations: Basic arithmetic operations such as addition, subtraction, multiplication, and division are used to simplify expressions and calculate the sum of the series.