Evaluate the Summation sum from k=1 to 4 of (-3/5)^k
The problem is asking you to calculate the sum of a finite series where each term is a successive power of -3/5, starting with the first power and ending with the fourth power. The summation notation provided describes the mathematical operation of adding together all the terms of this geometric sequence with a common ratio of -3/5. Each term in the series will be a power of -3/5, specifically -3/5 to the first power, -3/5 to the second power, and so on until -3/5 to the fourth power. The final result should be the value obtained after performing this summation.
$\sum_{k = 1}^{4} \left(\left(\right. - \frac{3}{5} \left.\right)\right)^{k}$
Answer
Solution:
Step 1: Write out the series
Write out the terms of the series for each value of $k$: $(-\frac{3}{5})^1 + (-\frac{3}{5})^2 + (-\frac{3}{5})^3 + (-\frac{3}{5})^4$.
Step 2: Simplify the series
Step 2.1: Simplify each term individually
Step 2.1.1: Simplify the first term
$-\frac{3}{5}$.
Step 2.1.2: Apply the exponentiation rule
Apply the rule $(ab)^n = a^n b^n$ to simplify the remaining terms.
Step 2.1.2.1: Simplify the second term
$(-1)^2(\frac{3}{5})^2$.
Step 2.1.2.2: Simplify the third term
$(-1)^3(\frac{3}{5})^3$.
Step 2.1.2.3: Simplify the fourth term
$(-1)^4(\frac{3}{5})^4$.
Step 2.2: Find a common denominator
Convert all terms to have a common denominator for easy addition.
Step 2.3: Combine the numerators
Combine the numerators over the common denominator.
Step 2.4: Simplify the fraction
Simplify the fraction by performing the arithmetic operations in the numerator.
Step 3: Present the result
The result can be expressed in its exact form and decimal form.
Exact Form: $-\frac{204}{625}$ Decimal Form: $-0.3264$
Knowledge Notes:
The problem involves evaluating a finite geometric series, which 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 power rule for exponents states that $(ab)^n = a^n b^n$, which allows us to simplify terms involving exponents.
When combining fractions, a common denominator is needed. The least common denominator is the least common multiple of the denominators.
Simplifying the fraction involves performing addition and subtraction on the numerators and keeping the common denominator.
The final result can be expressed in exact form (as a fraction) or in decimal form, which is a rounded or approximate value of the fraction.
