Sure! To write an explicit rule for this geometric sequence, let's follow a detailed, step-by-step process:
1. Identify the first term:
The first term of the sequence (a₁) is given as 10.
2. Find the common ratio:
The common ratio (r) is the factor by which each term is multiplied to get the next term. We can find it by dividing any term by its preceding term:
[tex]\[
r = \frac{a_2}{a_1} = \frac{40}{10} = 4
\][/tex]
This confirms that the common ratio is 4.
3. Write the general formula for the nth term of a geometric sequence:
The general formula for the nth term [tex]\( a_n \)[/tex] of a geometric sequence is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
4. Substitute the identified values into the formula:
Now, substituting the first term [tex]\( a_1 = 10 \)[/tex] and the common ratio [tex]\( r = 4 \)[/tex] into the general formula, we get:
[tex]\[
a_n = 10 \cdot 4^{(n-1)}
\][/tex]
So, the explicit rule for the given geometric sequence is:
[tex]\[
a_n = 10 \cdot 4^{(n-1)}
\][/tex]
