Problem 62 - Algebra and Trigonometry

Question

In Exercises \(57-62,\) let $$ \begin{array}{l} {\left\\{a_{n}\right\\}=-5,10,-20,40, \ldots} \\ {\left\\{b_{n}\right\\}=10,-5,-20,-35, \ldots} \end{array} $$ and $$\left\\{c_{n}\right\\}=-2,1,-\frac{1}{2}, \frac{1}{4}, \ldots$$ Find the product of the sum of the first 9 terms of \(\left\\{a_{n}\right\\}\) and the sum of the infinite series containing all the terms of \(\left\\{c_{n}\right\\}\)

Step-by-Step Solution

Answer

The product of the sum of the first 9 terms of sequence \(\left\{a_{n}\right\}\) and the sum of the infinite series \(\left\{c_{n}\right\}\) is 5100.

Step 1: Calculate the sum of the first 9 terms of \(\left\{a_{n}\right\}\)

The sequence \(\left\{a_{n}\right\}\) is a geometric sequence with the first term \(a_1 = -5\) and the common ratio \(r = -2\). We can use the formula for the sum of the first \(n\) terms of a geometric sequence, which is \[S_n = a_1 \frac{1 - r^n}{1 - r}\]. Substituting the given values, the sum of the first 9 terms of \(\left\{a_{n}\right\}\) is \[S_9 = -5 \frac{1 - (-2)^9}{1 - -2} = -1275\].

Step 2: Calculate the sum of the infinite series \(\left\{c_{n}\right\}\)

The sequence \(\left\{c_{n}\right\}\) is also a geometric sequence with the first term \(a_1 = -2\) and the common ratio \(r = -0.5\). We can use the formula for the sum of an infinite geometric series, which is \[S = \frac{a_1}{1 - r}\]. Substituting the given values, the sum of the infinite series \(\left\{c_{n}\right\}\) is \[S = \frac{-2}{1 - -0.5} = -4\].

Step 3: Find the product of the sums

Now that we have the sums of \(\left\{a_{n}\right\}\) and \(\left\{c_{n}\right\}\), we can find the product of these sums. The final result, denoted by \(P\), can be calculated as \(P = S_9 \times S = -1275 \times -4 = 5100\).

Key Concepts

Sum of Sequences

Understanding the sum of a sequence, particularly a geometric sequence, is crucial for solving many mathematical problems. A sequence is a set of numbers following a specific order. In a geometric sequence, each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. When asked to find the sum of the first 'n' terms, it involves adding up all those terms together. The formula for sum of the first 'n' terms of a geometric sequence is \[S_n = a_1 \frac{1 - r^n}{1 - r}\], where:

• \(a_1\) is the first term.
• \(r\) is the common ratio.
• \(n\) is the total number of terms you are adding up.

By applying this formula, you get a single number that represents the total of that sequence over the specified number of terms.

Infinite Series

Infinite series are sequences that continue indefinitely. Unlike finite sequences with a clearly defined endpoint, infinite series theoretically go on forever. In mathematics, when we work with infinite series, we're often interested in calculating the sum, especially if the series converges, meaning the sequence gets closer and closer to a specific number. For a geometric sequence with an absolute value of the common ratio less than 1, the infinite series has a sum that can be calculated using \[S = \frac{a_1}{1 - r}\] where:

• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio, with \(|r| < 1\).

This formula helps us find a finite sum of an otherwise infinitely long series.

Common Ratio

The common ratio is a central concept in geometric sequences. It is what you multiply each term in the sequence by to get the next term. This value remains constant throughout the sequence. Recognizing the common ratio is essential as it determines how rapidly the terms increase or decrease. For example, in a sequence like \(-5, 10, -20, 40, \ldots\), you identify that multiplying by \(-2\) yields each subsequent term, thus \(-2\) is the common ratio here. In another sequence such as \(-2, 1, -\frac{1}{2}, \frac{1}{4}, \ldots\), each term results from multiplying by \(-0.5\), making that the common ratio. Understanding and calculating this ratio reveals the dynamics behind the sequence's progression.

Geometric Series Formula

The geometric series formula is a valuable tool for finding the sum of terms in a geometric sequence. There are two main formulas:

• The formula for the sum of the first 'n' terms: \[S_n = a_1 \frac{1 - r^n}{1 - r}\]
• The infinite series formula, used when \(|r| < 1\): \[S = \frac{a_1}{1 - r}\]

These formulas allow us to solve problems involving geometric sequences efficiently. Suppose you need to find the sum of an infinite geometric sequence. Knowing the first term \(a_1\) and the common ratio \(r\), you can directly apply the appropriate formula. This easy approach is particularly beneficial in examining sequences that trend towards zero, providing a finite sum despite having infinite terms. Mastery of the geometric series formula turns complex arithmetic tasks into simple calculations.