Problem 58 - 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 \(a_{11}+b_{11}\)

Step-by-Step Solution

Answer

After calculation, the sum of the 11th terms of the sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) is: \(a_{11}+b_{11}= -5 \cdot (-2)^{10} + 10 - 50\).

Step 1: Find the 11th term of the geometric sequence

A geometric sequence is defined by \(a_{n}=a_{1} \cdot r^{(n-1)}\), where \(a_{1}\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. Substituting \(a_{1} = -5\), \(r = -2\), and \(n = 11\) into the equation, we obtain the 11th term of \(\{a_{n}\}\): \(a_{11}= -5 \cdot (-2)^{11-1}\).

Step 2: Find the 11th term of the arithmetic sequence

An arithmetic sequence is defined by \(b_{n} = b_{1}+ (n-1)d\), where \(b_{1}\) is the first term, \(d\) is the common difference, and \(n\) is the term number. Substituting \(b_{1} = 10\), \(d = -5\), and \(n = 11\) into the equation, we obtain the 11th term of \(\{b_{n}\}\): \(b_{11} = 10 + 10 \cdot (-5)\).

Step 3: Add the two terms together

Now we simply need to add together the 11th terms of the two sequences \(a_{11}\) and \(b_{11}\): \(a_{11}+b_{11} = (-5 \cdot (-2)^{11-1})+ (10 + 10 \cdot -5)\).

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. In simple terms, it builds a pattern through multiplication.
Consider the sequence \( \{a_n\} = -5, 10, -20, 40, \ldots \). To identify it as a geometric sequence, observe the ratios between consecutive terms. Here, the ratio

• -20 divided by 10 is\(-2\)
• 10 divided by -5 is \(-2\)

This common ratio (\(r\)) of \(-2\) confirms that \(\{a_n\}\) is a geometric sequence. Each term is created by multiplying the previous one by \(-2\). This multiplier, the common ratio, is the secret sauce of geometric sequences, allowing us to find any term using the formula:\(a_n = a_1 \cdot r^{(n-1)}\).
Sequences like this play a crucial role when looking for patterns in exponential growth and decay, making them applicable in finance, science, and nature.

Arithmetic Sequence

An arithmetic sequence, in contrast, is a sequence where each term is created by adding or subtracting the same value, known as the common difference, to the previous term. This pattern is built through addition or subtraction.
Let's examine the sequence \( \{b_n\} = 10, -5, -20, -35, \ldots \). Observe the differences between consecutive terms. Here

• -5 minus 10 is \(-15\)
• -20 minus (-5) is also \(-15\)

This common difference (\(d\)) of \(-15\) indicates that \(\{b_n\}\) is an arithmetic sequence. Every term is simply \(-15\) less than the term before it. The general formula for finding the nth term in such a sequence is \(b_n = b_1 + (n-1)d\).
Arithmetic sequences are useful in describing repetitive or linear processes, such as time intervals or equally spaced events.

Common Difference

The common difference in an arithmetic sequence is the value added to move from one term to the next. If this value is positive, the sequence increases, and if negative, it decreases.
In the sequence \( \{b_n\} = 10, -5, -20, -35, \ldots \), the common difference, \(d\), is \(-15\). This tells us two things:

• The sequence is decreasing.
• Each term reduces by \(15\) from the previous one.

Such a consistent pattern offers predictability, making it easier to calculate terms further along in the sequence. Using the formula \(b_n = b_1 + (n-1)d\), we can find any term in the sequence by adjusting technically only two components: the first term and the common difference.
Understanding the common difference aids not only in calculating terms but also in comprehending the sequence's progression and pattern nature.

Common Ratio

The common ratio is a defining feature of a geometric sequence, as it represents the fixed value by which each term is multiplied to reach the next one. Unlike the arithmetic sequence, which adds or subtracts, the common ratio involves multiplication or division.
In \( \{a_n\} = -5, 10, -20, 40, \ldots \), this ratio is \(-2\). This indicates that to get from one term to the next, each term multiplies by \(-2\).
As seen in our example, because \(\{-5 \rightarrow 10, 10 \rightarrow -20, -20 \rightarrow 40\}\), every step is consistently multiplied by \(-2\). This ratio provides a predictable increase or decrease, commonly found in scenarios of exponential growth. It's worth noting that the common ratio greatly impacts the pattern; a negative ratio, like this one, results in oscillating terms between positive and negative values.
The formula \(a_n = a_1 \cdot r^{(n-1)}\), utilizing the common ratio, ensures every term is accurately obtainable, highlighting its central role in defining a geometric sequence's structure.