Problem 11

Question

Find the $n$th term, the fifth term, and the eighth term of the geometric sequence. $$1,-x^{2}, x^{4},-x^{6}, \ldots$$

Step-by-Step Solution

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Answer

The formula for the nth term is $(-x^2)^{n-1}$; the fifth term is $x^8$, and the eighth term is $-x^{14}$.

1Step 1: Identify the first term and the common ratio

The first term of the sequence is 1. The second term is $-x^2$. To find the common ratio $r$, divide the second term by the first term: \[r = \frac{-x^2}{1} = -x^2\]. Therefore, the common ratio $r$ is $-x^2$.

2Step 2: Use the formula for the nth term of a geometric sequence

The formula for the nth term $a_n$ of a geometric sequence with the first term $a_1$ and common ratio $r$ is $a_n = a_1 \times r^{(n-1)}$. Substitute the known values: $a_1 = 1$ and $r = -x^2$. Thus, $a_n = 1 \times (-x^2)^{(n-1)} = (-x^2)^{(n-1)}$.

3Step 3: Find the fifth term

Substitute $n = 5$ into the nth term formula to find the fifth term: \[a_5 = (-x^2)^{(5-1)} = (-x^2)^4\]. Simplifying, $(-x^2)^4 = (x^8)$, as we square $-x^2$ two times, which makes it positive.

4Step 4: Find the eighth term

Substitute $n = 8$ into the nth term formula to find the eighth term: \[a_8 = (-x^2)^{(8-1)} = (-x^2)^7\]. Simplifying, $(-x^2)^7 = -x^{14}$, which follows from the rules of exponents where $(-1)^7 = -1$ and $(x^2)^7 = x^{14}$.

Key Concepts

nth term formulacommon ratioterms of a sequence

nth term formula

In a geometric sequence, each term is derived by multiplying the previous term by a constant value known as the common ratio. To find the value of any term in the sequence, we use the nth term formula:

Given the first term, denoted as $a_1$.
The common ratio is represented by $r$.
The position of the term we want to find is $n$.

Using these, the formula to find the nth term is: \[a_n = a_1 \times r^{(n-1)}\]This formula is very helpful because it allows us to find any term in the sequence quickly and without having to list all previous terms. It essentially gives us a framework to jump directly to any place in the sequence.

common ratio

The common ratio is a key component of a geometric sequence. This ratio stays the same between consecutive terms in the sequence. To find the common ratio, you simply divide any term in the sequence by the term immediately preceding it.

In the sequence, $1, -x^2, x^4, \ldots$
The formula to compute the common ratio is $r = \frac{a_2}{a_1}$.
Here, $a_2 = -x^2$ and $a_1 = 1$.

After computing, we find that the common ratio $r = -x^2$. This negative sign indicates that each term alternates in sign as the sequence progresses. Understanding the common ratio helps predict how each subsequent term behaves, which is crucial for grasping the nature of geometric sequences.

terms of a sequence

In a geometric sequence, the pattern follows a specific multiplicative progression defined by the first term and the common ratio. For the sequence here, starting with the first term 1, each next term is the previous term multiplied by the common ratio $-x^2$.
This sequence has the first few terms:

The first term: $1$
Second term: $-x^2$
Third term: $x^4$
The process continues in this multiplicative pattern.

The exercise particularly asked for the 5th term and 8th term, which can be calculated using the nth term formula: