Problem 61 - 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 6 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 6 terms of the sequence \(\left\{a_{n}\right\}\) and the sum of the infinite series of the sequence \(\left\{c_{n}\right\}\) is -140.

Step 1: Sum of the first 6 terms of sequence \(\left\{a_{n}\right\}\)

This sequence alternates signs and each term is two times the absolute value of the previous term. Thus, we calculate: \(-5 + 10 - 20 + 40 -80 + 160 = 105\)

Step 2: Sum of the infinite series for sequence \(\left\{c_{n}\right\}\)

This is a geometric series where the first term \(a = -2\) and the common ratio \(r = -1/2\). The sum of an infinite geometric series \(\sum a*r^n\) is given by \(S = a / (1-r)\) when \(-1 < r < 1\). Here, \(-1 < -1/2 < 1\), so we can plug \(a\) and \(r\) into the formula: \(S = -2 / (1 - (-1/2)) = -2 / (3/2) = -4/3\).

Step 3: Find the product of the two sums

The product of the two sum is \(105*(-4/3) = -140\).

Key Concepts

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Unlike a finite series that has a limited number of terms, an infinite series continues indefinitely and is often represented as an infinite sum.
In mathematics, infinite series can converge or diverge. A series converges if the partial sums approach a specific number, while it diverges if they do not. When dealing with a geometric series, it may converge if the absolute value of the common ratio is less than one.
For example, in the sequence \(\{c_{n}\}\) from the exercise, the terms create an infinite geometric series. Since the common ratio is \(-1/2\), which lies between \(-1\) and \(1\), it converges. This characteristic allows us to find a finite sum for the infinite sequence using the formula:

• \(S = \frac{a}{1 - r}\)

where \(a\) is the first term and \(r\) is the common ratio. Using this formula, the sum of the sequence \(\{c_{n}\}\) was calculated as \(-\frac{4}{3}\).
Understanding infinite series and whether they converge or diverge is crucial in many areas of calculus and mathematical analysis.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. Unlike a geometric series, which multiplies by a common ratio, an arithmetic sequence adds a fixed number to each term to get to the next. This difference is known as the "common difference".
A classic example can be visualized as:

• \(a_1, a_1 + d, a_1 + 2d, \ldots\)

where \(d\) represents the common difference.
Though the exercise specifically examines geometric and alternating series, understanding arithmetic sequences helps differentiate between these mathematical progressions. For instance, sequence \(\{b_{n}\}\) seems misleadingly similar to an arithmetic sequence initially, but upon deeper analysis, the sequence defies having a consistent common difference. This illustration underscores that not all sequences fit neatly into arithmetic or geometric categories without close examination.
Recognizing the pattern of arithmetic sequences can simplify many mathematical problems by giving a straightforward formula to calculate specific terms or the sequence’s sum.

Alternating Series

An alternating series is a series that alternates in sign with each term. This means that after a positive term, the next will be negative, and this pattern continues throughout the series.
Alternating series is crucial, especially in calculus, because its convergence can sometimes be determined using the Alternating Series Test. Essentially, for a series \(\sum (-1)^n a_n\), where the \(a_n\) terms are positive, the series converges if two conditions hold:

• The terms \(a_n\) are decreasing in magnitude.
• The limit of \(a_n\) as \(n\) approaches infinity is zero.

In the original problem, the sequence \(\{a_{n}\}\) exhibits an alternating pattern where the signs switch from negative to positive. In addition, each term magnifies by twice the absolute value, showcasing exponential growth. Calculating alternating series requires precise attention to detail because of the sign changes, influencing the total sum considerably.
By breaking down the alternating characteristics of sequences, learners can more effectively determine the sum or behavior of complex series problems.