Evaluate the Summation sum from x=2 to 12 of 4(-3)^(x-1)
The problem presented is a mathematical expression that requires evaluating a summation (specifically, a finite series). The summation is defined with the variable x starting at 2 and going up to 12. For each value of x, you are supposed to calculate the term 4(-3)^(x-1), which involves raising negative three to the power of (x-1) and then multiplying the result by 4. After calculating the term for each integer value of x between 2 and 12 inclusive, you would then sum up all these resulting values to get the final answer. This is a task in discrete mathematics and calculus, involving exponentiation and summation of a series.
$\sum_{x = 2}^{12} 4 \left(\left(\right. - 3 \left.\right)\right)^{x - 1}$
Answer
Solution:
Step:1
To calculate the sum of a finite geometric series, use 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.
Step:2
Determine the common ratio ($r$) by using the formula $r = \frac{a_{x+1}}{a_{x}}$ and perform the necessary substitutions.
Step:2.1
Insert $a_{x}$ and $a_{x+1}$ into the ratio formula: $r = \frac{4(-3)^{x}}{4(-3)^{x-1}}$.
Step:2.2
Proceed to simplify the expression.
Step:2.2.1
Eliminate the common factor of $4$: $r = \frac{\cancel{4}(-3)^{x}}{\cancel{4}(-3)^{x-1}}$.
Step:2.2.2
Reduce the powers of $(-3)$ by factoring out $(-3)^{x-1}$: $r = \frac{(-3)^{x-1}(-3)^{2-x}}{(-3)^{x-1}}$.
Step:2.2.3
Cancel out the common base and exponents: $r = \frac{\cancel{(-3)^{x-1}}(-3)^{2-x}}{\cancel{(-3)^{x-1}}}$.
Step:2.2.4
The expression simplifies to $r = (-3)^{2-x}$.
Step:2.2.5
Solve for $r$ by evaluating the exponent: $r = (-3)^{1}$.
Step:2.2.6
Conclude that the common ratio is $r = -3$.
Step:3
Find the first term ($a$) of the series by substituting $x=2$ into the given expression $4(-3)^{x-1}$.
Step:3.1
Substitute $2$ for $x$: $a = 4(-3)^{2-1}$.
Step:3.2
Simplify to find the first term: $a = 4(-3)^{1} = 4 \cdot -3 = -12$.
Step:4
Insert the values for $a$, $r$, and the number of terms ($n=12-2+1=11$) into the summation formula: $S = -12 \left(\frac{1 - (-3)^{11}}{1 - (-3)}\right)$.
Step:5
Simplify the expression to find the sum.
Step:5.1
Work on the numerator: $S = -12 \left(\frac{1 - (-3)^{11}}{1 - (-3)}\right) = -12 \left(\frac{1 + 177147}{1 - (-3)}\right)$.
Step:5.2
Simplify the denominator: $S = -12 \left(\frac{177148}{4}\right)$.
Step:5.3
Reduce the fraction by canceling out the common factor of $4$: $S = -3 \cdot 177148$.
Step:5.4
Complete the calculation: $S = -531444$.
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 common ratio ($r$).
Sum of a Finite Geometric Series: The sum of the first $n$ terms of a geometric series can be calculated using 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.
Simplification of Exponents: When simplifying expressions with exponents, remember that dividing powers with the same base can be done by subtracting the exponents: $\frac{a^{m}}{a^{n}} = a^{m-n}$.
Arithmetic Operations: Basic arithmetic operations such as addition, subtraction, multiplication, and division are used to simplify expressions.
Negative Exponents: A negative exponent means that the base is reciprocal, and the negative sign indicates the reciprocal of the base raised to the absolute value of the exponent: $a^{-n} = \frac{1}{a^{n}}$.
Evaluating Powers: When evaluating an expression with a power, the exponent indicates how many times the base is multiplied by itself.
Simplifying Fractions: To simplify a fraction, look for common factors in the numerator and denominator that can be canceled out.
Distributive Property: The distributive property states that $a(b + c) = ab + ac$. This is often used in algebra to multiply a single term and two or more terms inside a set of parentheses.
