Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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]
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]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.