Problem 30 - Algebra and Trigonometry

Question

Find the indicated term of each geometric sequence. 10th term of \(-1,2,-4, \ldots\)

Step-by-Step Solution

Answer

The 10th term is 512.

Step 1: Identify the first term and common ratio

The first term of the geometric sequence is given as onumber\(a_1 = -1otitleotag\). We need to find the common ratio, which is the factor by which we multiply each term to get to the next term. Using the first two terms: onumber\(\frac{2}{-1} = -2otitleotag\), the common ratio (r) is onumber\(r = -2otitleotag\).

Step 2: Use the formula for the n-th term of a geometric sequence

The formula for the n-th term of a geometric sequence is given by onumber\(a_n = a_1 \times r^{(n-1)}otitleotag\). Substituting the values we have: onumber\(a_{10} = -1 \times (-2)^{10-1}otitleotag\).

Step 3: Calculate the 10th term

Now calculate onumber\((-2)^{9}otitleotag\). This equals onumber\(-512otitleotag\). Therefore, onumber\(a_{10} = -1 \times -512 = 512otitleotag\).

Step 4: Write the final answer

The 10th term of the geometric sequence is onumber\(512otitleotag\).

Key Concepts

First Term in Geometric Sequences

The first term of a geometric sequence is the initial value from which the sequence begins. It is often denoted as \(a_1\). In our exercise, the first term \(a_1\) is given as \(-1\). This term is crucial because it sets the stage for the entire sequence. Every subsequent term is derived based on this initial value.
Understanding the first term helps us to see where the sequence starts and makes it easier to work with the formula for finding other terms in the sequence.

Common Ratio in Geometric Sequences

The common ratio, often denoted by \(r\), is the factor by which each term of the sequence is multiplied to get the next term. It can be found by dividing the second term by the first term of the sequence. In our exercise, the first two terms are \(-1\) and \(2\). So, the common ratio is:
\(r = \frac{2}{-1} = -2\)
The common ratio is crucial because it determines how the sequence progresses. If the ratio is positive, the terms will stay either all positive or all negative. If the ratio is negative, like in our case, the signs of the terms will alternate. This alternating nature is evident from the sequence: \(-1, 2, -4, \text{...}\).

Finding the N-th Term of a Geometric Sequence

The formula for finding the n-th term of a geometric sequence is given by:
\(a_n = a_1 \times r^{(n-1)}\)
This formula lets us find any term in the sequence if we know the first term \(a_1\) and the common ratio \(r\). In our exercise, we are asked to find the 10th term. Substituting the values for \(a_1 = -1\) and \(r = -2\), we use:\
10th term formula calculation:
\(a_{10} = -1 \times (-2)^{10-1}\)
This simplifies to \(a_{10} = -1 \times (-2)^9\). We calculate \((-2)^9\) which equals \(-512\). So, the term \(a_{10}\) is:
\(a_{10} = -1 \times -512 = 512\)
Thus, the 10th term is \(512\). This formula is powerful because it allows us to compute any term in the sequence without having to list all previous terms.