Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Find the sum. $$\sum_{k=1}^{7}\left(3^{-k}\right)$$

Step-by-Step Solution

Answer

The sum is \(\frac{1093}{2187}\).

Step 1: Identify the Series Terms

The given series is a sum of the terms from a geometric sequence: \(3^{-k}\) from \(k=1\) to \(k=7\). The terms are: \(3^{-1}, 3^{-2}, 3^{-3}, 3^{-4}, 3^{-5}, 3^{-6}, 3^{-7}\).

Step 2: Determine the First Term and Common Ratio

In a geometric series, the first term \(a\) is \(3^{-1} = \frac{1}{3}\), and the common ratio \(r\) is \(3^{-1} = \frac{1}{3}\), since each term is obtained by multiplying the previous term by \(\frac{1}{3}\).

Step 3: Use the Geometric Series Sum Formula

The formula for the sum of the first \(n\) terms of a geometric series is given by: \[S_n = a \frac{1-r^n}{1-r}\]where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Here, \(n=7\).

Step 4: Plug Values into the Formula

Substitute \(a = \frac{1}{3}\), \(r = \frac{1}{3}\), and \(n = 7\) into the sum formula: \[S_7 = \frac{1}{3} \frac{1-(\frac{1}{3})^7}{1-\frac{1}{3}}\]

Step 5: Simplify the Denominator

Calculate the denominator:\[1 - \frac{1}{3} = \frac{2}{3}\]

Step 6: Simplify the Numerator

Calculate the numerator:\[1 - (\frac{1}{3})^7 = 1 - \frac{1}{2187} = \frac{2186}{2187}\]

Step 7: Calculate the Series Sum

Substitute the simplified numerator and denominator back into the formula:\[S_7 = \frac{1}{3} \times \frac{2186}{2187} \times \frac{3}{2}\]Thus,\[S_7 = \frac{2186}{4374}\]

Step 8: Simplify the Fraction

Simplify the fraction \(\frac{2186}{4374}\) to its simplest form, which is \(\frac{1093}{2187}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2.

Key Concepts

Geometric Sequence

A geometric sequence, often referred to as a geometric progression, is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. In simpler terms, you can think of it as a multiplied chain of numbers. The common ratio can be a positive or negative fraction or a whole number.
For instance, in the case of the series given, the terms are derived from such a sequence: \(3^{-1}, 3^{-2}, 3^{-3}, 3^{-4}, 3^{-5}, 3^{-6}, 3^{-7}\). Notice how each term is the preceding term multiplied by \(\frac{1}{3}\). This characteristic is the essence of a geometric sequence. The knowledge of geometric sequences is essential for understanding their applications in various mathematical and real-life problems such as finance (compound interest) and population growth models.

• The first term in this sequence is \(3^{-1} = \frac{1}{3}\).
• Each term can be obtained by multiplying the previous term by the common ratio \(r = \frac{1}{3}\).

Understanding geometric sequences paves the way for solving more complex problems involving geometric series.

Sum Formula for Geometric Series

The sum of a geometric series can be quickly calculated using a convenient formula. This formula saves you the time and effort of manually adding terms. When you know a few key details, like the first term, common ratio, and the number of terms, the formula gives you the sum instantly.
The formula for the sum of the first \(n\) terms of a geometric series is:
\[ S_n = a \frac{1-r^n}{1-r} \]
where:

• \(S_n\) is the sum of the series.
• \(a\) is the first term of the series.
• \(r\) is the common ratio.
• \(n\) is the total number of terms.

The application of this formula is straightforward. Consider the given series where \(a = \frac{1}{3}\), \(r = \frac{1}{3}\), and \(n = 7\). Plug them into the formula to find the sum of the series.
Using this formula effectively requires careful calculation of both the numerator and the denominator terms, as they involve powers and fractions.

Fraction Simplification

Fraction simplification is an essential skill in mathematics. It involves reducing fractions to their lowest terms for a simplified expression, making calculations and interpretations easier. This is done by dividing both the numerator and denominator by their greatest common divisor (GCD).
In the exercise, after plugging values into the geometric series sum formula, the result was \(\frac{2186}{4374}\). Although that's correct, it's not as simple as it could be. To simplify it:

1. Find the GCD of the numerator (2186) and the denominator (4374), which is 2 in this case.
2. Divide both the numerator and the denominator by this GCD.

The calculation gives \(\frac{1093}{2187}\), a much simplified version. Recognizing when and how to simplify fractions ensures accuracy and makes further calculations more manageable.

Common Ratio

The common ratio is a fundamental characteristic of a geometric sequence. It's the constant factor between consecutive terms. Understanding it helps you predict future terms in the sequence or even sum them up easily.
In any geometric sequence, the common ratio \(r\) can be calculated by dividing any term by the previous term. For the given series, you've seen that:

• Starting from the second term, each term is \( \frac{1}{3} \) of the previous one.
• Mathematically, this is noted by the fact that \(r = \frac{1}{3}\).

Knowing the common ratio helps identify the behavior of the sequence. If \(|r| < 1\), the terms get smaller, approaching zero, which is typical in decay processes like radioactive decay or financial depreciation models. Conversely, if \(|r| > 1\), terms increase quickly, modeling exponential growth scenarios.
The common ratio remains a straightforward yet powerful concept in the arsenal of mathematical tools.