Find the nth term of the geometric sequence: -2, ~6_ -18- -54 ~162
The Correct Answer and Explanation is:
To find the nth term of a geometric sequence, we use the formula:an=a⋅rn−1a_n = a \cdot r^{n-1}an=a⋅rn−1
Where:
- ana_nan is the nth term
- aaa is the first term
- rrr is the common ratio
- nnn is the term number
Step 1: Identify the first term and common ratio
From the given sequence:−2, 6, −18, 54, −162, …-2,\ 6,\ -18,\ 54,\ -162,\ \ldots−2, 6, −18, 54, −162, …
- The first term a=−2a = -2a=−2
- To find the common ratio, divide the second term by the first:
r=6−2=−3r = \frac{6}{-2} = -3r=−26=−3
So each term is multiplied by -3 to get the next one.
Step 2: Plug into the formula
Now substitute into the geometric sequence formula:an=−2⋅(−3)n−1a_n = -2 \cdot (-3)^{n-1}an=−2⋅(−3)n−1
Final Answer:
an=−2(−3)n−1\boxed{a_n = -2(-3)^{n-1}}an=−2(−3)n−1
Explanation (300 words):
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed number known as the common ratio. In this problem, we are given a sequence: -2, 6, -18, 54, -162. To find a formula that represents the nth term of this sequence, we begin by identifying key components.
The first term is simply the starting number of the sequence, which is -2. This value is referred to as aaa in the general formula.
The common ratio is calculated by dividing any term by the one that comes before it. For instance, 6 divided by -2 gives -3. Repeating this with other terms confirms the ratio is consistent: -18 divided by 6 also equals -3, and so on. This confirms the sequence is indeed geometric, and the common ratio rrr is -3.
Using the geometric formula an=a⋅rn−1a_n = a \cdot r^{n-1}an=a⋅rn−1, we substitute the known values. This gives us an=−2⋅(−3)n−1a_n = -2 \cdot (-3)^{n-1}an=−2⋅(−3)n−1. This expression will correctly generate every term in the sequence. For example, for n=1n = 1n=1, a1=−2⋅(−3)0=−2a_1 = -2 \cdot (-3)^0 = -2a1=−2⋅(−3)0=−2. For n=2n = 2n=2, a2=−2⋅(−3)1=6a_2 = -2 \cdot (-3)^1 = 6a2=−2⋅(−3)1=6, and so on.
This formula is useful because it lets you find any term in the sequence without listing all the terms before it.
