Evaluate the Summation sum from j=1 to 4 of 54(1/3)^j
The given problem is asking for the calculation of a finite mathematical series. Specifically, the series is the sum of a sequence of terms that are given by a formula involving an exponential function with a base of 54 and an exponent that decreases by a factor of 1/3 for each sequential value of j, starting with j=1 and ending with j=4. The task is to evaluate the sum of all the terms in this series when j takes on the integer values 1, 2, 3, and 4.
$\sum_{j = 1}^{4} 54 \left(\left(\right. \frac{1}{3} \left.\right)\right)^{j}$
Answer
Solution:
Step 1: Write out the series for each value of $j$.
$54 \left(\frac{1}{3}\right)^1 + 54 \left(\frac{1}{3}\right)^2 + 54 \left(\frac{1}{3}\right)^3 + 54 \left(\frac{1}{3}\right)^4$
Step 2: Begin simplification.
Step 2.1: Simplify each term individually.
Step 2.1.1: Simplify the first term.
$54 \left(\frac{1}{3}\right) + 54 \left(\frac{1}{3}\right)^2 + 54 \left(\frac{1}{3}\right)^3 + 54 \left(\frac{1}{3}\right)^4$
Step 2.1.2: Reduce by common factors.
Step 2.1.2.1: Extract the factor of 3 from 54.
$3 \cdot 18 \cdot \frac{1}{3} + 54 \left(\frac{1}{3}\right)^2 + 54 \left(\frac{1}{3}\right)^3 + 54 \left(\frac{1}{3}\right)^4$
Step 2.1.2.2: Simplify the common factors.
$18 + 54 \left(\frac{1}{3}\right)^2 + 54 \left(\frac{1}{3}\right)^3 + 54 \left(\frac{1}{3}\right)^4$
Step 2.1.3: Apply exponent rules.
$18 + 54 \cdot \frac{1}{3^2} + 54 \left(\frac{1}{3}\right)^3 + 54 \left(\frac{1}{3}\right)^4$
Step 2.1.4: Continue simplification.
Step 2.1.4.1: Simplify the second term.
$18 + 54 \cdot \frac{1}{9} + 54 \left(\frac{1}{3}\right)^3 + 54 \left(\frac{1}{3}\right)^4$
Step 2.1.4.2: Reduce by common factors.
$18 + 6 + 54 \left(\frac{1}{3}\right)^3 + 54 \left(\frac{1}{3}\right)^4$
Step 2.1.5: Repeat the process for the third term.
$18 + 6 + 54 \cdot \frac{1}{3^3} + 54 \left(\frac{1}{3}\right)^4$
Step 2.1.6: Simplify the third term.
$18 + 6 + 2 + 54 \left(\frac{1}{3}\right)^4$
Step 2.1.7: Finally, simplify the fourth term.
$18 + 6 + 2 + 54 \cdot \frac{1}{3^4}$
Step 2.1.8: Simplify the fourth term.
$18 + 6 + 2 + \frac{2}{3}$
Step 2.2: Combine terms over a common denominator.
$18 + 6 + 2 + \frac{2}{3}$
Step 2.3: Convert all terms to fractions.
$\frac{54}{3} + \frac{18}{3} + \frac{6}{3} + \frac{2}{3}$
Step 2.4: Add the numerators.
$\frac{54 + 18 + 6 + 2}{3}$
Step 2.5: Simplify the sum.
$\frac{80}{3}$
Step 3: Present the result in various forms.
Exact Form: $\frac{80}{3}$ Decimal Form: $26.\overline{6}$ Mixed Number Form: $26 \frac{2}{3}$
Knowledge Notes:
This problem involves evaluating a finite geometric series. The steps taken to solve the problem include expanding the series, simplifying each term, and combining like terms. The following knowledge points are relevant:
Geometric Series: 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.
Exponent Rules: When raising a fraction to a power, both the numerator and the denominator are raised to that power.
Simplifying Fractions: Factors common to the numerator and denominator can be canceled out.
Combining Like Terms: When adding fractions, they must have a common denominator. Whole numbers can be converted to fractions by using 1 as the denominator.
Mixed Numbers: A mixed number is a whole number and a proper fraction combined, such as $26 \frac{2}{3}$.
Understanding these concepts is essential for solving problems involving summation and series.
