Evaluate the Summation sum from k=1 to 8 of 10(-2/3)^k
The question is asking to compute the total sum of a series where each term is defined by the expression 10(-2/3)^k. The series starts from k=1 and continues until k=8. Each term is a product of 10 and a power of -2/3, with k increasing by 1 for each subsequent term. The problem requires you to find the combined value of the terms when the value of k ranges from 1 to 8.
$\sum_{k = 1}^{8} 10 \left(\left(\right. - \frac{2}{3} \left.\right)\right)^{k}$
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 $S$ is the sum, $a$ 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 $r = \frac{a_{k+1}}{a_k}$.
Step:2.1
Insert the values for $a_{k}$ and $a_{k+1}$ into the ratio formula: $r = \frac{10(-\frac{2}{3})^{k+1}}{10(-\frac{2}{3})^k}$.
Step:2.2
Proceed to simplify the expression.
Step:2.2.1
Eliminate the common factor of $10$: $r = \frac{\cancel{10}(-\frac{2}{3})^{k+1}}{\cancel{10}(-\frac{2}{3})^k}$.
Step:2.2.2
Further simplify by canceling out the common term $(-\frac{2}{3})^k$: $r = \frac{(-\frac{2}{3})^{k}(-\frac{2}{3})}{(-\frac{2}{3})^k}$.
Step:2.2.2.1
Isolate $(-\frac{2}{3})^k$ from $(-\frac{2}{3})^{k+1}$: $r = \frac{(-\frac{2}{3})^k(-\frac{2}{3})}{(-\frac{2}{3})^k}$.
Step:2.2.2.2
Cancel out the common factors: $r = \frac{\cancel{(-\frac{2}{3})^k}(-\frac{2}{3})}{\cancel{(-\frac{2}{3})^k}}$.
Step:2.2.2.2.3
Simplify the ratio to $r = -\frac{2}{3}$.
Step:3
Identify the first term of the series by setting $k=1$: $a = 10(-\frac{2}{3})^1$.
Step:3.2
Simplify the first term: $a = 10(-\frac{2}{3})$.
Step:3.2.2
Perform the multiplication: $a = -\frac{20}{3}$.
Step:4
Insert the values of $r$, $a$, and $n$ into the sum formula: $S = -\frac{20}{3} \left( \frac{1 - (-\frac{2}{3})^8}{1 - (-\frac{2}{3})} \right)$.
Step:5
Simplify the expression to find the sum.
Step:5.1
Work on the numerator: $1 - (-\frac{2}{3})^8$.
Step:5.1.1
Distribute the exponent: $1 - ((-1)^8(\frac{2}{3})^8)$.
Step:5.1.2
Simplify the negative exponent: $1 - ((-1)^8 \cdot -1 \cdot \frac{2^8}{3^8})$.
Step:5.1.3
Calculate the negative power of $-1$: $1 - \frac{2^8}{3^8}$.
Step:5.1.4
Compute the powers of $2$ and $3$: $1 - \frac{256}{6561}$.
Step:5.1.7
Combine the fractions: $-\frac{20}{3} \left( \frac{6561 - 256}{6561} \right)$.
Step:5.2
Simplify the denominator: $1 - (-\frac{2}{3})$.
Step:5.3
Multiply the numerator by the reciprocal of the denominator: $-\frac{20}{3} \left( \frac{6305}{6561} \cdot \frac{3}{5} \right)$.
Step:5.6
Multiply the fractions: $-\frac{25220}{6561}$.
Step:6
The final sum can be expressed in various forms:
Exact Form: $-\frac{25220}{6561}$ Decimal Form: Approximately $-3.8439$ Mixed Number Form: $-3 \frac{5537}{6561}$
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.
Sum of 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.
Common Ratio: In a geometric series, the common ratio $r$ is the factor by which each term is multiplied to get the next term. It can be found by dividing any term by the previous term: $r = \frac{a_{k+1}}{a_k}$.
Simplifying Expressions: When simplifying expressions, look for common factors in the numerator and denominator that can be canceled out. Also, use exponent rules such as $a^m \cdot a^n = a^{m+n}$ and $(ab)^n = a^n b^n$.
Negative Exponents: A negative exponent means that the base is reciprocated and then raised to the absolute value of the exponent: $a^{-n} = \frac{1}{a^n}$.
Fraction Operations: When adding, subtracting, multiplying, or dividing fractions, it is often necessary to find a common denominator or to multiply by the reciprocal.
