To find the explicit formula for the geometric sequence given by the terms [tex]\(4, 20, 100, 500, \ldots\)[/tex], follow these steps:
### Step 1: Identify the First Term
The first term of the sequence is:
[tex]\[ a_1 = 4 \][/tex]
### Step 2: Determine the Common Ratio
To find the common ratio ([tex]\(r\)[/tex]), divide the second term by the first term:
[tex]\[ r = \frac{\text{second term}}{\text{first term}} = \frac{20}{4} = 5 \][/tex]
We can double-check this by also dividing the third term by the second term:
[tex]\[ r = \frac{100}{20} = 5 \][/tex]
And from the fourth term by the third term:
[tex]\[ r = \frac{500}{100} = 5 \][/tex]
So, the common ratio is consistently [tex]\(5\)[/tex].
### Step 3: Write the Explicit Formula
The explicit formula for a geometric sequence is given by:
[tex]\[ a(n) = a_1 \cdot r^{(n-1)} \][/tex]
Where:
- [tex]\( a(n) \)[/tex] is the [tex]\( n \)[/tex]-th term,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the term number.
Substitute the values we have:
[tex]\[ a(n) = 4 \cdot 5^{(n-1)} \][/tex]
### Step 4: Simplify the Formula
We can simplify the notation by expressing the formula in exponential form as follows:
[tex]\[ a(n) = 4 \cdot 5^x \][/tex]
where [tex]\( x \)[/tex] can represent [tex]\( n-1 \)[/tex].
### Step 5: Verify Against the Given Options
Among the given options:
- [tex]\( a(n) = 4 + 16x \)[/tex]
- [tex]\( a(n) = 4 - 16x \)[/tex]
- [tex]\( a(n) = 4 \left( \frac{1}{5} \right)^x \)[/tex]
- [tex]\( a(n) = 4 \cdot 5^x \)[/tex]
The correct formula for the geometric sequence is:
[tex]\[ a(n) = 4 \cdot 5^x \][/tex]
Thus, the explicit formula for the sequence is:
[tex]\[ \boxed{a(n) = 4(5)^x} \][/tex]
