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Sagot :
Sure! Let's break down the problem and find the solution step by step.
### Part (i): Finding the sum [tex]\( S \)[/tex] of the first [tex]\( n \)[/tex] terms of the sequence whose [tex]\( r \)[/tex]-th term is [tex]\( 4 \cdot 2^r \)[/tex]
The sequence given is:
[tex]\[ a_r = 4 \cdot 2^r \][/tex]
We recognize that this sequence is a geometric sequence where each term is multiplied by a common ratio. To write it in a standard form, consider the first few terms:
- When [tex]\( r = 1 \)[/tex],
[tex]\[ a_1 = 4 \cdot 2^1 = 8 \][/tex]
- When [tex]\( r = 2 \)[/tex],
[tex]\[ a_2 = 4 \cdot 2^2 = 16 \][/tex]
- When [tex]\( r = 3 \)[/tex],
[tex]\[ a_3 = 4 \cdot 2^3 = 32 \][/tex]
Thus, the general term can be written as:
[tex]\[ a_r = 4 \cdot 2^{r-1} \][/tex]
The sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric sequence is given by:
[tex]\[ S_n = a_1 \cdot \frac{r^n - 1}{r - 1} \][/tex]
In our sequence:
- The first term [tex]\( a_1 = 8 \)[/tex] (since [tex]\( 4 \cdot 2^1 \)[/tex])
- The common ratio [tex]\( r = 2 \)[/tex]
Substituting these values into the formula, we get:
[tex]\[ S_n = 8 \cdot \frac{2^n - 1}{2 - 1} = 8 \cdot (2^n - 1) \][/tex]
For example, let's take [tex]\( n = 10 \)[/tex]:
[tex]\[ S_{10} = 8 \cdot (2^{10} - 1) \][/tex]
[tex]\[ S_{10} = 8 \cdot (1024 - 1) \][/tex]
[tex]\[ S_{10} = 8 \cdot 1023 \][/tex]
[tex]\[ S_{10} = 8184 \][/tex]
### Part (ii): Finding the value of [tex]\( n \)[/tex] for which the difference between [tex]\( S \)[/tex] and 4 is less than 104
We need to find [tex]\( n \)[/tex] such that:
[tex]\[ |S_n - 4| < 104 \][/tex]
From Part (i), we have:
[tex]\[ S_n = 8 \cdot (2^n - 1) \][/tex]
We want the difference [tex]\( |S_n - 4| \)[/tex] to be less than 104:
[tex]\[ |8 \cdot (2^n - 1) - 4| < 104 \][/tex]
Let's simplify the expression inside the absolute value:
[tex]\[ 8 \cdot (2^n - 1) - 4 = 8 \cdot 2^n - 8 - 4 = 8 \cdot 2^n - 12 \][/tex]
Thus, the condition becomes:
[tex]\[ |8 \cdot 2^n - 12| < 104 \][/tex]
Solving for the inequality:
[tex]\[ 8 \cdot 2^n - 12 < 104 \][/tex]
[tex]\[ 8 \cdot 2^n - 12 > -104 \][/tex]
First inequality:
[tex]\[ 8 \cdot 2^n < 116 \][/tex]
[tex]\[ 2^n < 14.5 \][/tex]
Taking the base 2 logarithm:
[tex]\[ n < \log_2 (14.5) \approx 3.857 \][/tex]
Since [tex]\( n \)[/tex] must be an integer:
[tex]\[ n \leq 3 \][/tex]
Second inequality:
[tex]\[ 8 \cdot 2^n > -92 \][/tex]
[tex]\[ 2^n > -11.5 \][/tex]
Since [tex]\( 2^n \)[/tex] is always positive, this inequality is always true.
Therefore, the integer value [tex]\( n = 1 \)[/tex] satisfies the condition where the difference between [tex]\( S_n \)[/tex] and 4 is less than 104.
This means the value of [tex]\( n \)[/tex] that satisfies the condition is [tex]\( 1 \)[/tex].
### Summary:
(i) The sum [tex]\( S \)[/tex] of the first 10 terms of the sequence is [tex]\( 4092 \)[/tex].
(ii) The value [tex]\( n \)[/tex] for which the difference between [tex]\( S \)[/tex] and 4 is less than 104 is [tex]\( 1 \)[/tex].
### Part (i): Finding the sum [tex]\( S \)[/tex] of the first [tex]\( n \)[/tex] terms of the sequence whose [tex]\( r \)[/tex]-th term is [tex]\( 4 \cdot 2^r \)[/tex]
The sequence given is:
[tex]\[ a_r = 4 \cdot 2^r \][/tex]
We recognize that this sequence is a geometric sequence where each term is multiplied by a common ratio. To write it in a standard form, consider the first few terms:
- When [tex]\( r = 1 \)[/tex],
[tex]\[ a_1 = 4 \cdot 2^1 = 8 \][/tex]
- When [tex]\( r = 2 \)[/tex],
[tex]\[ a_2 = 4 \cdot 2^2 = 16 \][/tex]
- When [tex]\( r = 3 \)[/tex],
[tex]\[ a_3 = 4 \cdot 2^3 = 32 \][/tex]
Thus, the general term can be written as:
[tex]\[ a_r = 4 \cdot 2^{r-1} \][/tex]
The sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric sequence is given by:
[tex]\[ S_n = a_1 \cdot \frac{r^n - 1}{r - 1} \][/tex]
In our sequence:
- The first term [tex]\( a_1 = 8 \)[/tex] (since [tex]\( 4 \cdot 2^1 \)[/tex])
- The common ratio [tex]\( r = 2 \)[/tex]
Substituting these values into the formula, we get:
[tex]\[ S_n = 8 \cdot \frac{2^n - 1}{2 - 1} = 8 \cdot (2^n - 1) \][/tex]
For example, let's take [tex]\( n = 10 \)[/tex]:
[tex]\[ S_{10} = 8 \cdot (2^{10} - 1) \][/tex]
[tex]\[ S_{10} = 8 \cdot (1024 - 1) \][/tex]
[tex]\[ S_{10} = 8 \cdot 1023 \][/tex]
[tex]\[ S_{10} = 8184 \][/tex]
### Part (ii): Finding the value of [tex]\( n \)[/tex] for which the difference between [tex]\( S \)[/tex] and 4 is less than 104
We need to find [tex]\( n \)[/tex] such that:
[tex]\[ |S_n - 4| < 104 \][/tex]
From Part (i), we have:
[tex]\[ S_n = 8 \cdot (2^n - 1) \][/tex]
We want the difference [tex]\( |S_n - 4| \)[/tex] to be less than 104:
[tex]\[ |8 \cdot (2^n - 1) - 4| < 104 \][/tex]
Let's simplify the expression inside the absolute value:
[tex]\[ 8 \cdot (2^n - 1) - 4 = 8 \cdot 2^n - 8 - 4 = 8 \cdot 2^n - 12 \][/tex]
Thus, the condition becomes:
[tex]\[ |8 \cdot 2^n - 12| < 104 \][/tex]
Solving for the inequality:
[tex]\[ 8 \cdot 2^n - 12 < 104 \][/tex]
[tex]\[ 8 \cdot 2^n - 12 > -104 \][/tex]
First inequality:
[tex]\[ 8 \cdot 2^n < 116 \][/tex]
[tex]\[ 2^n < 14.5 \][/tex]
Taking the base 2 logarithm:
[tex]\[ n < \log_2 (14.5) \approx 3.857 \][/tex]
Since [tex]\( n \)[/tex] must be an integer:
[tex]\[ n \leq 3 \][/tex]
Second inequality:
[tex]\[ 8 \cdot 2^n > -92 \][/tex]
[tex]\[ 2^n > -11.5 \][/tex]
Since [tex]\( 2^n \)[/tex] is always positive, this inequality is always true.
Therefore, the integer value [tex]\( n = 1 \)[/tex] satisfies the condition where the difference between [tex]\( S_n \)[/tex] and 4 is less than 104.
This means the value of [tex]\( n \)[/tex] that satisfies the condition is [tex]\( 1 \)[/tex].
### Summary:
(i) The sum [tex]\( S \)[/tex] of the first 10 terms of the sequence is [tex]\( 4092 \)[/tex].
(ii) The value [tex]\( n \)[/tex] for which the difference between [tex]\( S \)[/tex] and 4 is less than 104 is [tex]\( 1 \)[/tex].
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