Answer :
Sure! Let's determine which formulas correctly represent the [tex]\(n^{\text{th}}\)[/tex] term of the sequence based on the given options.
Given the sequence definitions and initial conditions, we want to see which ones successfully calculate the same [tex]\(n^{\text{th}}\)[/tex] term.
#### First Option
[tex]\[a_n = 20000 + 20000(n-1)\][/tex]
To see if this works, let's check for [tex]\(n = 3\)[/tex]:
[tex]\[a_3 = 20000 + 20000(3-1) = 20000 + 20000 \times 2 = 20000 + 40000 = 60000\][/tex]
This option is correct since it gives us the same result as calculated.
#### Second Option
[tex]\[
\begin{array}{c}
a_1 = 20000 \\
a_n = a_{n-1} + 20000
\end{array}
\][/tex]
For [tex]\(n = 3\)[/tex]:
- First term: [tex]\(a_1 = 20000\)[/tex]
- Second term: [tex]\(a_2 = a_1 + 20000 = 20000 + 20000 = 40000\)[/tex]
- Third term: [tex]\(a_3 = a_2 + 20000 = 40000 + 20000 = 60000\)[/tex]
This option is also correct because it results in [tex]\( 60000 \)[/tex].
#### Third Option
[tex]\[
\begin{array}{c}
a_n = 20000 + 0(n-1) \\
a_n = 20000 n + 20000 \\
a_n = 20000n
\end{array}
\][/tex]
Let's break down these individually:
- [tex]\(a_n = 20000 + 0(n-1)\)[/tex]:
For any [tex]\(n\)[/tex], it simplifies to:
[tex]\[a_n = 20000\][/tex]
This does not match our sequence for higher values of [tex]\(n\)[/tex], for example, [tex]\(n = 3\)[/tex] results in [tex]\(a_3 = 20000\)[/tex], which is not correct for our sequence.
- [tex]\(a_n = 20000n + 20000\)[/tex]:
For [tex]\(n = 3\)[/tex]:
[tex]\[a_3 = 20000 \times 3 + 20000 = 60000 + 20000 = 80000\][/tex]
This result is clearly not correct.
- [tex]\(a_n = 20000n\)[/tex]:
For [tex]\(n = 3\)[/tex]:
[tex]\[a_3 = 20000 \times 3 = 60000\][/tex]
This option is correct because it gives us the same result.
### Conclusion
The correct formulas that represent the [tex]\(n^{\mathrm{th}}\)[/tex] term of the sequence are:
1. [tex]\(a_n = 20000 + 20000(n-1)\)[/tex]
2.
[tex]\[
\begin{array}{c}
a_1 = 20000 \\
a_n = a_{n-1} + 20000
\end{array}
\][/tex]
3. [tex]\(a_n = 20000n\)[/tex]
These options correctly produce the same [tex]\(n^{\text{th}}\)[/tex] term value.
Given the sequence definitions and initial conditions, we want to see which ones successfully calculate the same [tex]\(n^{\text{th}}\)[/tex] term.
#### First Option
[tex]\[a_n = 20000 + 20000(n-1)\][/tex]
To see if this works, let's check for [tex]\(n = 3\)[/tex]:
[tex]\[a_3 = 20000 + 20000(3-1) = 20000 + 20000 \times 2 = 20000 + 40000 = 60000\][/tex]
This option is correct since it gives us the same result as calculated.
#### Second Option
[tex]\[
\begin{array}{c}
a_1 = 20000 \\
a_n = a_{n-1} + 20000
\end{array}
\][/tex]
For [tex]\(n = 3\)[/tex]:
- First term: [tex]\(a_1 = 20000\)[/tex]
- Second term: [tex]\(a_2 = a_1 + 20000 = 20000 + 20000 = 40000\)[/tex]
- Third term: [tex]\(a_3 = a_2 + 20000 = 40000 + 20000 = 60000\)[/tex]
This option is also correct because it results in [tex]\( 60000 \)[/tex].
#### Third Option
[tex]\[
\begin{array}{c}
a_n = 20000 + 0(n-1) \\
a_n = 20000 n + 20000 \\
a_n = 20000n
\end{array}
\][/tex]
Let's break down these individually:
- [tex]\(a_n = 20000 + 0(n-1)\)[/tex]:
For any [tex]\(n\)[/tex], it simplifies to:
[tex]\[a_n = 20000\][/tex]
This does not match our sequence for higher values of [tex]\(n\)[/tex], for example, [tex]\(n = 3\)[/tex] results in [tex]\(a_3 = 20000\)[/tex], which is not correct for our sequence.
- [tex]\(a_n = 20000n + 20000\)[/tex]:
For [tex]\(n = 3\)[/tex]:
[tex]\[a_3 = 20000 \times 3 + 20000 = 60000 + 20000 = 80000\][/tex]
This result is clearly not correct.
- [tex]\(a_n = 20000n\)[/tex]:
For [tex]\(n = 3\)[/tex]:
[tex]\[a_3 = 20000 \times 3 = 60000\][/tex]
This option is correct because it gives us the same result.
### Conclusion
The correct formulas that represent the [tex]\(n^{\mathrm{th}}\)[/tex] term of the sequence are:
1. [tex]\(a_n = 20000 + 20000(n-1)\)[/tex]
2.
[tex]\[
\begin{array}{c}
a_1 = 20000 \\
a_n = a_{n-1} + 20000
\end{array}
\][/tex]
3. [tex]\(a_n = 20000n\)[/tex]
These options correctly produce the same [tex]\(n^{\text{th}}\)[/tex] term value.
