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Sagot :
Let's first define the general terms for both the arithmetic sequence and the geometric sequence based on the information provided in the problem.
### Arithmetic Sequence
The first term, [tex]\(a_1\)[/tex], is 400, and the common difference, [tex]\(d\)[/tex], is 500. The [tex]\(n\)[/tex]th term of an arithmetic sequence can be expressed as:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Substituting the given values:
[tex]\[ a_n = 400 + (n-1) \cdot 500 \][/tex]
[tex]\[ a_n = 400 + 500n - 500 \][/tex]
[tex]\[ a_n = 500n - 100 \][/tex]
### Geometric Sequence
The first term, [tex]\(g_1\)[/tex], is 3, and the common ratio, [tex]\(r\)[/tex], is 3. The [tex]\(n\)[/tex]th term of a geometric sequence can be expressed as:
[tex]\[ g_n = g_1 \cdot r^{n-1} \][/tex]
Substituting the given values:
[tex]\[ g_n = 3 \cdot 3^{n-1} \][/tex]
[tex]\[ g_n = 3^n \][/tex]
### Finding [tex]\(n\)[/tex] such that [tex]\( g_n > a_n \)[/tex]
We need to find the smallest [tex]\(n\)[/tex] for which the [tex]\(n\)[/tex]th term of the geometric sequence exceeds the [tex]\(n\)[/tex]th term of the arithmetic sequence, i.e., [tex]\( g_n > a_n \)[/tex].
Set the expressions for [tex]\(a_n\)[/tex] and [tex]\(g_n\)[/tex] and solve for [tex]\(n\)[/tex]:
[tex]\[ 3^n > 500n - 100 \][/tex]
### Evaluating terms
Let's evaluate the sequences term by term until we find when the geometric sequence term first exceeds the arithmetic sequence term:
1. For [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 500 \cdot 1 - 100 = 400 \][/tex]
[tex]\[ g_1 = 3^1 = 3 \][/tex]
Clearly, [tex]\(3 < 400\)[/tex].
2. For [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 500 \cdot 2 - 100 = 900 \][/tex]
[tex]\[ g_2 = 3^2 = 9 \][/tex]
Clearly, [tex]\(9 < 900\)[/tex].
3. For [tex]\(n = 3\)[/tex]:
[tex]\[ a_3 = 500 \cdot 3 - 100 = 1400 \][/tex]
[tex]\[ g_3 = 3^3 = 27 \][/tex]
Clearly, [tex]\(27 < 1400\)[/tex].
4. For [tex]\(n = 4\)[/tex]:
[tex]\[ a_4 = 500 \cdot 4 - 100 = 1900 \][/tex]
[tex]\[ g_4 = 3^4 = 81 \][/tex]
Clearly, [tex]\(81 < 1900\)[/tex].
5. For [tex]\(n = 5\)[/tex]:
[tex]\[ a_5 = 500 \cdot 5 - 100 = 2400 \][/tex]
[tex]\[ g_5 = 3^5 = 243 \][/tex]
Clearly, [tex]\(243 < 2400\)[/tex].
6. For [tex]\(n = 6\)[/tex]:
[tex]\[ a_6 = 500 \cdot 6 - 100 = 2900 \][/tex]
[tex]\[ g_6 = 3^6 = 729 \][/tex]
Clearly, [tex]\(729 < 2900\)[/tex].
7. For [tex]\(n = 7\)[/tex]:
[tex]\[ a_7 = 500 \cdot 7 - 100 = 3400 \][/tex]
[tex]\[ g_7 = 3^7 = 2187 \][/tex]
Clearly, [tex]\(2187 < 3400\)[/tex].
8. For [tex]\(n = 8\)[/tex]:
[tex]\[ a_8 = 500 \cdot 8 - 100 = 3900 \][/tex]
[tex]\[ g_8 = 3^8 = 6561 \][/tex]
Here, [tex]\(6561 > 3900\)[/tex].
So, the least number [tex]\(n\)[/tex] such that the [tex]\(n\)[/tex]th term of the geometric sequence is greater than the corresponding term in the arithmetic sequence is:
[tex]\[ n = 8 \][/tex]
Therefore, the geometric sequence is larger than the arithmetic sequence at the [tex]\(8\)[/tex]th term.
### Arithmetic Sequence
The first term, [tex]\(a_1\)[/tex], is 400, and the common difference, [tex]\(d\)[/tex], is 500. The [tex]\(n\)[/tex]th term of an arithmetic sequence can be expressed as:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Substituting the given values:
[tex]\[ a_n = 400 + (n-1) \cdot 500 \][/tex]
[tex]\[ a_n = 400 + 500n - 500 \][/tex]
[tex]\[ a_n = 500n - 100 \][/tex]
### Geometric Sequence
The first term, [tex]\(g_1\)[/tex], is 3, and the common ratio, [tex]\(r\)[/tex], is 3. The [tex]\(n\)[/tex]th term of a geometric sequence can be expressed as:
[tex]\[ g_n = g_1 \cdot r^{n-1} \][/tex]
Substituting the given values:
[tex]\[ g_n = 3 \cdot 3^{n-1} \][/tex]
[tex]\[ g_n = 3^n \][/tex]
### Finding [tex]\(n\)[/tex] such that [tex]\( g_n > a_n \)[/tex]
We need to find the smallest [tex]\(n\)[/tex] for which the [tex]\(n\)[/tex]th term of the geometric sequence exceeds the [tex]\(n\)[/tex]th term of the arithmetic sequence, i.e., [tex]\( g_n > a_n \)[/tex].
Set the expressions for [tex]\(a_n\)[/tex] and [tex]\(g_n\)[/tex] and solve for [tex]\(n\)[/tex]:
[tex]\[ 3^n > 500n - 100 \][/tex]
### Evaluating terms
Let's evaluate the sequences term by term until we find when the geometric sequence term first exceeds the arithmetic sequence term:
1. For [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 500 \cdot 1 - 100 = 400 \][/tex]
[tex]\[ g_1 = 3^1 = 3 \][/tex]
Clearly, [tex]\(3 < 400\)[/tex].
2. For [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 500 \cdot 2 - 100 = 900 \][/tex]
[tex]\[ g_2 = 3^2 = 9 \][/tex]
Clearly, [tex]\(9 < 900\)[/tex].
3. For [tex]\(n = 3\)[/tex]:
[tex]\[ a_3 = 500 \cdot 3 - 100 = 1400 \][/tex]
[tex]\[ g_3 = 3^3 = 27 \][/tex]
Clearly, [tex]\(27 < 1400\)[/tex].
4. For [tex]\(n = 4\)[/tex]:
[tex]\[ a_4 = 500 \cdot 4 - 100 = 1900 \][/tex]
[tex]\[ g_4 = 3^4 = 81 \][/tex]
Clearly, [tex]\(81 < 1900\)[/tex].
5. For [tex]\(n = 5\)[/tex]:
[tex]\[ a_5 = 500 \cdot 5 - 100 = 2400 \][/tex]
[tex]\[ g_5 = 3^5 = 243 \][/tex]
Clearly, [tex]\(243 < 2400\)[/tex].
6. For [tex]\(n = 6\)[/tex]:
[tex]\[ a_6 = 500 \cdot 6 - 100 = 2900 \][/tex]
[tex]\[ g_6 = 3^6 = 729 \][/tex]
Clearly, [tex]\(729 < 2900\)[/tex].
7. For [tex]\(n = 7\)[/tex]:
[tex]\[ a_7 = 500 \cdot 7 - 100 = 3400 \][/tex]
[tex]\[ g_7 = 3^7 = 2187 \][/tex]
Clearly, [tex]\(2187 < 3400\)[/tex].
8. For [tex]\(n = 8\)[/tex]:
[tex]\[ a_8 = 500 \cdot 8 - 100 = 3900 \][/tex]
[tex]\[ g_8 = 3^8 = 6561 \][/tex]
Here, [tex]\(6561 > 3900\)[/tex].
So, the least number [tex]\(n\)[/tex] such that the [tex]\(n\)[/tex]th term of the geometric sequence is greater than the corresponding term in the arithmetic sequence is:
[tex]\[ n = 8 \][/tex]
Therefore, the geometric sequence is larger than the arithmetic sequence at the [tex]\(8\)[/tex]th term.
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