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Sagot :
Sure! Let's solve each of the parts step by step for the given geometric sequences.
### Part a:
Given:
- First term, [tex]\( a_1 = 16 \)[/tex]
- Common ratio, [tex]\( r = \frac{1}{2} \)[/tex]
- Number of terms, [tex]\( n = 7 \)[/tex]
We need to find the sum [tex]\( S_n \)[/tex] of the first 7 terms of this geometric series.
The formula to calculate the sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric sequence is:
[tex]\[ S_n = a_1 \frac{1 - r^n}{1 - r} \][/tex]
Substitute the given values:
[tex]\[ S_7 = 16 \cdot \frac{1 - \left(\frac{1}{2}\right)^7}{1 - \left(\frac{1}{2}\right)} \][/tex]
Therefore, the sum [tex]\( S_7 \)[/tex] is:
[tex]\[ S_7 = 31.75 \][/tex]
### Part b:
Given:
- First term, [tex]\( a_1 = 4 \)[/tex]
- Last term, [tex]\( a_n = 256 \)[/tex]
- Common ratio, [tex]\( r = 4 \)[/tex]
We need to find the sum [tex]\( S_n \)[/tex] of the terms up to [tex]\( a_n = 256 \)[/tex], but first we need to determine the number of terms [tex]\( n \)[/tex].
The general term [tex]\( a_n \)[/tex] of a geometric sequence is given by:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
[tex]\[ 256 = 4 \cdot 4^{n-1} \][/tex]
Divide both sides by 4:
[tex]\[ 64 = 4^{n-1} \][/tex]
Since [tex]\( 64 = 4^3 \)[/tex], we get:
[tex]\[ 4^3 = 4^{n-1} \][/tex]
Thus:
[tex]\[ n - 1 = 3 \][/tex]
[tex]\[ n = 4 \][/tex]
Now we can find [tex]\( S_n \)[/tex]:
[tex]\[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \][/tex]
Substitute the values:
[tex]\[ S_4 = 4 \cdot \frac{1 - 4^4}{1 - 4} \][/tex]
Therefore, the sum [tex]\( S_4 \)[/tex] is:
[tex]\[ S_4 = 340.0 \][/tex]
### Part c:
Given:
- First term, [tex]\( a_1 = 1 \)[/tex]
- Fifth term, [tex]\( a_5 = \frac{1}{16} \)[/tex]
- Common ratio, [tex]\( r = -\frac{1}{2} \)[/tex]
We need to find the sum [tex]\( S_n \)[/tex] of the first 5 terms.
First, verify the common ratio:
[tex]\[ a_5 = a_1 \cdot r^{4} \][/tex]
[tex]\[ \frac{1}{16} = 1 \cdot \left(-\frac{1}{2}\right)^4 \][/tex]
[tex]\[ \frac{1}{16} = \frac{1}{16} \][/tex]
Thus, the common ratio [tex]\( r \)[/tex] is correct.
Now calculate [tex]\( S_n \)[/tex]:
[tex]\[ S_5 = a_1 \cdot \frac{1 - r^5}{1 - r} \][/tex]
Substitute the values:
[tex]\[ S_5 = 1 \cdot \frac{1 - \left(-\frac{1}{2}\right)^5}{1 - \left(-\frac{1}{2}\right)} \][/tex]
Therefore, the sum [tex]\( S_5 \)[/tex] is:
[tex]\[ S_5 = 0.6875 \][/tex]
### Part d:
Given:
- Sum of the first 4 terms, [tex]\( S_4 = 30 \)[/tex]
- Common ratio, [tex]\( r = -2 \)[/tex]
- Number of terms, [tex]\( n = 4 \)[/tex]
We need to find the first term [tex]\( a_1 \)[/tex] and the first three terms of the sequence.
The formula for [tex]\( S_n \)[/tex] is:
[tex]\[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \][/tex]
Substitute the given values:
[tex]\[ 30 = a_1 \cdot \frac{1 - (-2)^4}{1 - (-2)} \][/tex]
[tex]\[ 30 = a_1 \cdot \frac{1 - 16}{1 + 2} \][/tex]
[tex]\[ 30 = a_1 \cdot \frac{-15}{3} \][/tex]
[tex]\[ 30 = a_1 \cdot (-5) \][/tex]
[tex]\[ a_1 = -6 \][/tex]
Now, find the first three terms of the sequence:
First term [tex]\( a_1 \)[/tex]:
[tex]\[ a_1 = -6 \][/tex]
Second term [tex]\( a_2 \)[/tex]:
[tex]\[ a_2 = a_1 \cdot r = -6 \cdot (-2) = 12 \][/tex]
Third term [tex]\( a_3 \)[/tex]:
[tex]\[ a_3 = a_2 \cdot r = 12 \cdot (-2) = -24 \][/tex]
Therefore, the first three terms of the sequence are:
[tex]\[ -6, 12, -24 \][/tex]
In summary:
- Part a: [tex]\( S_7 = 31.75 \)[/tex]
- Part b: [tex]\( S_4 = 340.0 \)[/tex]
- Part c: [tex]\( S_5 = 0.6875 \)[/tex]
- Part d: The first three terms are [tex]\( -6, 12, -24 \)[/tex]
### Part a:
Given:
- First term, [tex]\( a_1 = 16 \)[/tex]
- Common ratio, [tex]\( r = \frac{1}{2} \)[/tex]
- Number of terms, [tex]\( n = 7 \)[/tex]
We need to find the sum [tex]\( S_n \)[/tex] of the first 7 terms of this geometric series.
The formula to calculate the sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric sequence is:
[tex]\[ S_n = a_1 \frac{1 - r^n}{1 - r} \][/tex]
Substitute the given values:
[tex]\[ S_7 = 16 \cdot \frac{1 - \left(\frac{1}{2}\right)^7}{1 - \left(\frac{1}{2}\right)} \][/tex]
Therefore, the sum [tex]\( S_7 \)[/tex] is:
[tex]\[ S_7 = 31.75 \][/tex]
### Part b:
Given:
- First term, [tex]\( a_1 = 4 \)[/tex]
- Last term, [tex]\( a_n = 256 \)[/tex]
- Common ratio, [tex]\( r = 4 \)[/tex]
We need to find the sum [tex]\( S_n \)[/tex] of the terms up to [tex]\( a_n = 256 \)[/tex], but first we need to determine the number of terms [tex]\( n \)[/tex].
The general term [tex]\( a_n \)[/tex] of a geometric sequence is given by:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
[tex]\[ 256 = 4 \cdot 4^{n-1} \][/tex]
Divide both sides by 4:
[tex]\[ 64 = 4^{n-1} \][/tex]
Since [tex]\( 64 = 4^3 \)[/tex], we get:
[tex]\[ 4^3 = 4^{n-1} \][/tex]
Thus:
[tex]\[ n - 1 = 3 \][/tex]
[tex]\[ n = 4 \][/tex]
Now we can find [tex]\( S_n \)[/tex]:
[tex]\[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \][/tex]
Substitute the values:
[tex]\[ S_4 = 4 \cdot \frac{1 - 4^4}{1 - 4} \][/tex]
Therefore, the sum [tex]\( S_4 \)[/tex] is:
[tex]\[ S_4 = 340.0 \][/tex]
### Part c:
Given:
- First term, [tex]\( a_1 = 1 \)[/tex]
- Fifth term, [tex]\( a_5 = \frac{1}{16} \)[/tex]
- Common ratio, [tex]\( r = -\frac{1}{2} \)[/tex]
We need to find the sum [tex]\( S_n \)[/tex] of the first 5 terms.
First, verify the common ratio:
[tex]\[ a_5 = a_1 \cdot r^{4} \][/tex]
[tex]\[ \frac{1}{16} = 1 \cdot \left(-\frac{1}{2}\right)^4 \][/tex]
[tex]\[ \frac{1}{16} = \frac{1}{16} \][/tex]
Thus, the common ratio [tex]\( r \)[/tex] is correct.
Now calculate [tex]\( S_n \)[/tex]:
[tex]\[ S_5 = a_1 \cdot \frac{1 - r^5}{1 - r} \][/tex]
Substitute the values:
[tex]\[ S_5 = 1 \cdot \frac{1 - \left(-\frac{1}{2}\right)^5}{1 - \left(-\frac{1}{2}\right)} \][/tex]
Therefore, the sum [tex]\( S_5 \)[/tex] is:
[tex]\[ S_5 = 0.6875 \][/tex]
### Part d:
Given:
- Sum of the first 4 terms, [tex]\( S_4 = 30 \)[/tex]
- Common ratio, [tex]\( r = -2 \)[/tex]
- Number of terms, [tex]\( n = 4 \)[/tex]
We need to find the first term [tex]\( a_1 \)[/tex] and the first three terms of the sequence.
The formula for [tex]\( S_n \)[/tex] is:
[tex]\[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \][/tex]
Substitute the given values:
[tex]\[ 30 = a_1 \cdot \frac{1 - (-2)^4}{1 - (-2)} \][/tex]
[tex]\[ 30 = a_1 \cdot \frac{1 - 16}{1 + 2} \][/tex]
[tex]\[ 30 = a_1 \cdot \frac{-15}{3} \][/tex]
[tex]\[ 30 = a_1 \cdot (-5) \][/tex]
[tex]\[ a_1 = -6 \][/tex]
Now, find the first three terms of the sequence:
First term [tex]\( a_1 \)[/tex]:
[tex]\[ a_1 = -6 \][/tex]
Second term [tex]\( a_2 \)[/tex]:
[tex]\[ a_2 = a_1 \cdot r = -6 \cdot (-2) = 12 \][/tex]
Third term [tex]\( a_3 \)[/tex]:
[tex]\[ a_3 = a_2 \cdot r = 12 \cdot (-2) = -24 \][/tex]
Therefore, the first three terms of the sequence are:
[tex]\[ -6, 12, -24 \][/tex]
In summary:
- Part a: [tex]\( S_7 = 31.75 \)[/tex]
- Part b: [tex]\( S_4 = 340.0 \)[/tex]
- Part c: [tex]\( S_5 = 0.6875 \)[/tex]
- Part d: The first three terms are [tex]\( -6, 12, -24 \)[/tex]
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