Answer :
Certainly! Let's break down how we arrived at the recursive formula for the given sequence step by step.
### Step-by-Step Solution:
1. Identify the Given Terms:
The sequence given is: [tex]\(5, 25, 125, 625, 3125, 15625, \ldots\)[/tex]
We are also given that the first term, [tex]\(a_1\)[/tex], is 5.
2. Observe the Pattern:
Let's look at how each term in the sequence is related to the previous term:
- [tex]\(a_1 = 5\)[/tex]
- [tex]\(a_2 = 25\)[/tex]
- [tex]\(a_3 = 125\)[/tex]
- [tex]\(a_4 = 625\)[/tex]
- [tex]\(a_5 = 3125\)[/tex]
- [tex]\(a_6 = 15625\)[/tex]
Notice that each term is obtained by multiplying the previous term by 5.
3. Determine the Relationship Between Consecutive Terms:
To get from [tex]\(a_1\)[/tex] to [tex]\(a_2\)[/tex]:
[tex]\[
25 = 5 \times 5
\][/tex]
To get from [tex]\(a_2\)[/tex] to [tex]\(a_3\)[/tex]:
[tex]\[
125 = 25 \times 5
\][/tex]
To get from [tex]\(a_3\)[/tex] to [tex]\(a_4\)[/tex]:
[tex]\[
625 = 125 \times 5
\][/tex]
The pattern continues similarly for further terms.
4. Form the Recursive Formula:
Based on the observed pattern, we can generalize the relationship for any term [tex]\(a_n\)[/tex]:
[tex]\[
a_n = 5 \times a_{n-1} \quad \text{for} \quad n > 1
\][/tex]
This means that each term [tex]\(a_n\)[/tex] is five times the previous term [tex]\(a_{n-1}\)[/tex].
### Final Recursive Formula:
Putting it all together, the recursive formula for the given sequence is:
[tex]\[
a_n = 5 \times a_{n-1} \quad \text{for} \quad n > 1
\][/tex]
Also, don't forget the initial term given:
[tex]\[
a_1 = 5
\][/tex]
This completes the step-by-step process of finding the recursive formula for the sequence.
### Step-by-Step Solution:
1. Identify the Given Terms:
The sequence given is: [tex]\(5, 25, 125, 625, 3125, 15625, \ldots\)[/tex]
We are also given that the first term, [tex]\(a_1\)[/tex], is 5.
2. Observe the Pattern:
Let's look at how each term in the sequence is related to the previous term:
- [tex]\(a_1 = 5\)[/tex]
- [tex]\(a_2 = 25\)[/tex]
- [tex]\(a_3 = 125\)[/tex]
- [tex]\(a_4 = 625\)[/tex]
- [tex]\(a_5 = 3125\)[/tex]
- [tex]\(a_6 = 15625\)[/tex]
Notice that each term is obtained by multiplying the previous term by 5.
3. Determine the Relationship Between Consecutive Terms:
To get from [tex]\(a_1\)[/tex] to [tex]\(a_2\)[/tex]:
[tex]\[
25 = 5 \times 5
\][/tex]
To get from [tex]\(a_2\)[/tex] to [tex]\(a_3\)[/tex]:
[tex]\[
125 = 25 \times 5
\][/tex]
To get from [tex]\(a_3\)[/tex] to [tex]\(a_4\)[/tex]:
[tex]\[
625 = 125 \times 5
\][/tex]
The pattern continues similarly for further terms.
4. Form the Recursive Formula:
Based on the observed pattern, we can generalize the relationship for any term [tex]\(a_n\)[/tex]:
[tex]\[
a_n = 5 \times a_{n-1} \quad \text{for} \quad n > 1
\][/tex]
This means that each term [tex]\(a_n\)[/tex] is five times the previous term [tex]\(a_{n-1}\)[/tex].
### Final Recursive Formula:
Putting it all together, the recursive formula for the given sequence is:
[tex]\[
a_n = 5 \times a_{n-1} \quad \text{for} \quad n > 1
\][/tex]
Also, don't forget the initial term given:
[tex]\[
a_1 = 5
\][/tex]
This completes the step-by-step process of finding the recursive formula for the sequence.
