Answer :
To find the explicit formula for the given geometric sequence, we need to identify the first term and the common ratio.
1. Identify the first term:
The sequence starts with 115. So, the first term [tex]\( a_1 \)[/tex] is 115.
2. Determine the common ratio:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio [tex]\( r \)[/tex].
Let's calculate the common ratio using the first two terms:
- First term ([tex]\( a_1 \)[/tex]) = 115
- Second term ([tex]\( a_2 \)[/tex]) = 690
To find the common ratio [tex]\( r \)[/tex], divide the second term by the first term:
[tex]\[
r = \frac{690}{115} = 6
\][/tex]
It's good practice to verify this ratio with other terms in the sequence. Let's check:
- Third term ([tex]\( a_3 \)[/tex]) = 4140
[tex]\[
\text{Ratio: } \frac{4140}{690} = 6
\][/tex]
- Fourth term ([tex]\( a_4 \)[/tex]) = 24840
[tex]\[
\text{Ratio: } \frac{24840}{4140} = 6
\][/tex]
Since the ratio is consistent across terms, the common ratio [tex]\( r \)[/tex] is 6.
3. Write the explicit formula:
The formula for the nth term of a geometric sequence is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Now, substituting the first term and the common ratio into the formula, we get:
[tex]\[
a_n = 115 \cdot 6^{(n-1)}
\][/tex]
Therefore, the explicit formula for this sequence is [tex]\( a_n = 115 \cdot (6)^{(n-1)} \)[/tex].
1. Identify the first term:
The sequence starts with 115. So, the first term [tex]\( a_1 \)[/tex] is 115.
2. Determine the common ratio:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio [tex]\( r \)[/tex].
Let's calculate the common ratio using the first two terms:
- First term ([tex]\( a_1 \)[/tex]) = 115
- Second term ([tex]\( a_2 \)[/tex]) = 690
To find the common ratio [tex]\( r \)[/tex], divide the second term by the first term:
[tex]\[
r = \frac{690}{115} = 6
\][/tex]
It's good practice to verify this ratio with other terms in the sequence. Let's check:
- Third term ([tex]\( a_3 \)[/tex]) = 4140
[tex]\[
\text{Ratio: } \frac{4140}{690} = 6
\][/tex]
- Fourth term ([tex]\( a_4 \)[/tex]) = 24840
[tex]\[
\text{Ratio: } \frac{24840}{4140} = 6
\][/tex]
Since the ratio is consistent across terms, the common ratio [tex]\( r \)[/tex] is 6.
3. Write the explicit formula:
The formula for the nth term of a geometric sequence is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Now, substituting the first term and the common ratio into the formula, we get:
[tex]\[
a_n = 115 \cdot 6^{(n-1)}
\][/tex]
Therefore, the explicit formula for this sequence is [tex]\( a_n = 115 \cdot (6)^{(n-1)} \)[/tex].