Problem 17 - Algebra for College Students

Question

The 10th term of \(1,-2,4,-8, \ldots\) \(-512\)

Step-by-Step Solution

Answer

The 10th term of the sequence is \(-512\).

Step 1: Identify the pattern

The given sequence is \(1, -2, 4, -8, \ldots\). This sequence alternates in signs and the magnitude grows exponentially. Observing the pattern, each term can be represented by multiplying the previous term by \(-2\). It resembles a geometric sequence.

Step 2: Write the general formula

In a geometric sequence, the general term \(a_n\) for the sequence with the first term \(a_1\) and common ratio \(r\) is given by \(a_n = a_1 \cdot r^{n-1}\). For this sequence, \(a_1 = 1\) and \(r = -2\). Therefore, the \(n\)-th term is given by \(a_n = 1 \cdot (-2)^{n-1}\).

Step 3: Calculate the 10th term

To find the 10th term, substitute \(n = 10\) into the general formula: \[a_{10} = 1 \cdot (-2)^{10-1} = (-2)^9\].

Step 4: Compute the power

Calculate \((-2)^9\). Since the exponent 9 is odd, the result will be negative. Calculate: \((-2)^9 = -512\). Therefore, the 10th term is \(-512\).

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases by the same factor over equal intervals of time. In the context of sequences, this concept means each term is obtained by multiplying the previous term by a constant, known as the common ratio. For example, the sequence

• 1, -2, 4, -8, ...

shows exponential growth because each term is the result of multiplying the previous term by -2. This type of growth is common in geometric sequences, where the larger the term number, the greater the effect of the exponent on the term's value.
Understanding exponential growth helps to predict future values in sequences quickly, especially when identifying patterns in multiplicative sequences.

Alternating Signs

Alternating signs refer to a pattern in a sequence where the signs of the terms change with each step. In the provided sequence, the terms alternate between positive and negative values:

• 1, -2, 4, -8, ...

This happens because the common ratio of the sequence is a negative number -2.
As a result, every time the term is multiplied by -2, the sign changes, introducing a simple but important pattern within the sequence. Alternating signs often indicate oscillating behavior, which is crucial in understanding sequences generated by negative common ratios.

Sequence Patterns

Recognizing patterns in sequences can simplify the process of finding specific terms and understanding the behavior of the sequence. For the sequence

• 1, -2, 4, -8, ...

we notice a repeating pattern in both the magnitude and the sign of the terms.
The term's magnitude follows a power of two, while the sign alternates between positive and negative due to the common ratio being -2. To generalize, understanding sequence patterns involves identifying whether the sequence is arithmetic or geometric, the impact of the common ratio or difference, and any other changes such as sign alterations.
This understanding allows us to construct formulas like \( a_n = 1 \cdot (-2)^{n-1} \)to describe all terms accurately.