Problem 54 - Algebra and Trigonometry

Question

Expand each sum. \(\sum_{k=0}^{n}\left(\frac{3}{2}\right)^{k}\)

Step-by-Step Solution

Answer

This series sums to 2\left(\left(\frac{3}{2}\right)^{n+1}-1\).

Step 1: Identify the Series Type

Recognize that the series \(\sum_{k=0}^{n}\left(\frac{3}{2}\right)^{k}\) is a geometric series.

Step 2: Write the General Form of a Geometric Series

A geometric series with first term \(a\) and common ratio \(r\) is given by \(p{k=0}^{n}ar^{k}\).

Step 3: Substitute the Values

Here, \(a = 1\) (since there is no multiplying constant) and \(r = \frac{3}{2}\). The series becomes \(p{k=0}{n}\left(\frac{3}{2}\right)^{k}\).

Step 4: Apply the Sum Formula for a Finite Geometric Series

The sum of the first \((n+1)\) terms of a geometric series is given by \(p{k=0}^{n}ar^{k} = a\frac{r^{n+1}-1}{r-1}\). Using \(a = 1\) and \(r = \frac{3}{2}\), the sum becomes \(\left(\frac{3}{2}\right)^{n+1}-1}{\frac{3}{2}-1}\).

Step 5: Simplify the Expression

Simplify the fraction to get \(p{k=0}^{n}\left(\frac{3}{2}\right)^{k} = \frac{\left(\frac{3}{2}\right)^{n+1}-1}{\frac{1}{2}}\), which simplifies further to \(p{k=0}^{n}\left(\frac{3}{2}\right)^{k} = 2\left(\left(\frac{3}{2}\right)^{n+1}-1\).

Key Concepts

Sum of Geometric Series

The sum of a geometric series is a fundamental concept in mathematics, especially in sequences and series. A geometric series is defined as a series where each term is a constant multiple of the previous term. This constant multiple is known as the common ratio and is denoted by \(r\). Given the first term \(a\) and the common ratio \(r\), the geometric series can be written as:
\[ a, ar, ar^2, ar^3, \text{...}, ar^n \]
The sum of the first \(n+1\) terms of a geometric series is found using the sum formula:
\[ S_n = a \frac{r^{n+1} - 1}{r - 1} \]
This formula provides a straightforward way to calculate the sum of a geometric series, provided you know the terms \(a\) and \(r\). This allows for quick and efficient computation without needing to add each term individually.

Finite Geometric Series

A finite geometric series is simply a geometric series that contains a limited number of terms. For instance, if we take the series \( \textstyle\frac{3}{2} \), \( \textstyle\frac{3^2}{2^2} \), \( \textstyle\frac{3^3}{2^3} \), up to \( \textstyle\frac{3^{n}}{2^{n}} \), we can denote this series compactly using summation notation as:
\( \textstyle\frac{3}{2} \), \( \textstyle\frac{3^2}{2^2} \), \( \textstyle\frac{3^3}{2^3} \), \( \textstyle\frac{3^{n}}{2^{n}} \)
To find the sum, substitute the first term \(a = 1\) and the common ratio \(r = \textstyle\frac{3}{2} \) into the finite geometric series sum formula:
\( S_n = a \frac{r^{n+1} - 1}{r - 1} \)
This results in:
\( S_n = 1 \frac{\textstyle\frac{3}{2} \,^{n+1} - 1}{\textstyle\frac{3}{2} - 1} \)
This further simplifies to:
\( S_n = 1 \frac{\textstyle\frac{3}{2} \,^{n+1} - 1}{\textstyle\frac{1}{2}} \)
Which simplifies further to:
\( S_n = 2 \big(\textstyle\frac{3}{2} \,^{n+1} - 1\big) \).

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. When working with the sum of a geometric series, simplification is crucial to making the formula easier to work with and understand. For example, in the series \( \textstyle\frac{3}{2} \), \( \textstyle\frac{3^2}{2^2} \), \( \textstyle\frac{3^3}{2^3} \) up to \( \textstyle\frac{3^{n}}{2^{n}} \), we derived the sum expression:
\[ S_n = 2 \bigg(\bigg(\textstyle\frac{3}{2}\bigg)^{n+1} - 1\bigg) \]
In algebraic simplification, every step carefully reduces the number of mathematical operations and terms.
Here are some tips for simplifying algebraic expressions:

• Combine like terms: simplify terms with the same base and exponent.
• Factor: rewrite the expression by factoring common terms.
• Cancel common factors: when fractions are involved, cancel common factors in the numerator and denominator.
• Use the distributive property: expand and then combine like terms when necessary.

By applying these techniques, the summarized equation can be more manageable and solvable.