At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
