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Sagot :
Sure, let's go through this one step at a time.
### Activity 2
We are given the general rule for the terms in the table as:
[tex]\[ T_n = d n^2 + 1 \][/tex]
The table provides us with specific term numbers and their corresponding values.
| Term number | 1 | 2 | 3 | 4 | 5 | 11 | 20 |
|-------------|---|---|---|---|------------|------------|------------|
| Value | 3 | 9 | 19| 33| A = 51 | B = 243 | C = 801 |
From this, we need to determine several values and answer questions step-by-step:
### 2.1 Determine the value of [tex]\( d \)[/tex]
Using the provided equation [tex]\( T_n = d n^2 + 1 \)[/tex] and the given values for [tex]\( T_1, T_2, \)[/tex] and [tex]\( T_3 \)[/tex]:
- For [tex]\( T_1 = 3 \)[/tex]:
[tex]\[ 3 = d(1^2) + 1 \implies d \cdot 1^2 + 1 = 3 \implies d + 1 = 3 \implies d = 2 \][/tex]
- For [tex]\( T_2 = 9 \)[/tex]:
[tex]\[ 9 = d(2^2) + 1 \implies d \cdot 4 + 1 = 9 \implies 4d + 1 = 9 \implies 4d = 8 \implies d = 2 \][/tex]
- For [tex]\( T_3 = 19 \)[/tex]:
[tex]\[ 19 = d(3^2) + 1 \implies d \cdot 9 + 1 = 19 \implies 9d + 1 = 19 \implies 9d = 18 \implies d = 2 \][/tex]
From all these calculations, the value of [tex]\( d \)[/tex] is:
[tex]\[ d = 2 \][/tex]
### 2.2 Find the values of [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex]
- For [tex]\( T_5 \)[/tex]:
[tex]\[ T_5 = d(5^2) + 1 = 2(25) + 1 = 50 + 1 = 51 \][/tex]
So, [tex]\( A = 51 \)[/tex].
- For [tex]\( T_{11} \)[/tex]:
[tex]\[ T_{11} = d(11^2) + 1 = 2(121) + 1 = 242 + 1 = 243 \][/tex]
So, [tex]\( B = 243 \)[/tex].
- For [tex]\( T_{20} \)[/tex]:
[tex]\[ T_{20} = d(20^2) + 1 = 2(400) + 1 = 800 + 1 = 801 \][/tex]
So, [tex]\( C = 801 \)[/tex].
### 2.3 Determine the 17th term
For [tex]\( T_{17} \)[/tex]:
[tex]\[ T_{17} = d(17^2) + 1 = 2(289) + 1 = 578 + 1 = 579 \][/tex]
So, the 17th term is:
[tex]\[ T_{17} = 579 \][/tex]
### 1.2 Calculate the 36th term
For [tex]\( T_{36} \)[/tex]:
[tex]\[ T_{36} = d(36^2) + 1 = 2(1296) + 1 = 2592 + 1 = 2593 \][/tex]
So, the 36th term is:
[tex]\[ T_{36} = 2593 \][/tex]
### 1.2.3 Which term will be equal to [tex]\(-241\)[/tex]?
We need to solve:
[tex]\[ T_n = -241 \][/tex]
[tex]\[ d \cdot n^2 + 1 = -241 \][/tex]
[tex]\[ 2 \cdot n^2 + 1 = -241 \][/tex]
[tex]\[ 2n^2 + 1 = -241 \][/tex]
[tex]\[ 2n^2 = -242 \][/tex]
[tex]\[ n^2 = -121 \][/tex]
Considering [tex]\( n^2 = -121 \)[/tex] and recognizing that [tex]\( n \)[/tex] must be a real number, there is no real solution for [tex]\( n \)[/tex] because the square of a real number cannot be negative. Therefore, no term in this sequence will be equal to -241 in the context of real numbers.
By following these steps, we've addressed the parts of the problem as a whole.
### Activity 2
We are given the general rule for the terms in the table as:
[tex]\[ T_n = d n^2 + 1 \][/tex]
The table provides us with specific term numbers and their corresponding values.
| Term number | 1 | 2 | 3 | 4 | 5 | 11 | 20 |
|-------------|---|---|---|---|------------|------------|------------|
| Value | 3 | 9 | 19| 33| A = 51 | B = 243 | C = 801 |
From this, we need to determine several values and answer questions step-by-step:
### 2.1 Determine the value of [tex]\( d \)[/tex]
Using the provided equation [tex]\( T_n = d n^2 + 1 \)[/tex] and the given values for [tex]\( T_1, T_2, \)[/tex] and [tex]\( T_3 \)[/tex]:
- For [tex]\( T_1 = 3 \)[/tex]:
[tex]\[ 3 = d(1^2) + 1 \implies d \cdot 1^2 + 1 = 3 \implies d + 1 = 3 \implies d = 2 \][/tex]
- For [tex]\( T_2 = 9 \)[/tex]:
[tex]\[ 9 = d(2^2) + 1 \implies d \cdot 4 + 1 = 9 \implies 4d + 1 = 9 \implies 4d = 8 \implies d = 2 \][/tex]
- For [tex]\( T_3 = 19 \)[/tex]:
[tex]\[ 19 = d(3^2) + 1 \implies d \cdot 9 + 1 = 19 \implies 9d + 1 = 19 \implies 9d = 18 \implies d = 2 \][/tex]
From all these calculations, the value of [tex]\( d \)[/tex] is:
[tex]\[ d = 2 \][/tex]
### 2.2 Find the values of [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex]
- For [tex]\( T_5 \)[/tex]:
[tex]\[ T_5 = d(5^2) + 1 = 2(25) + 1 = 50 + 1 = 51 \][/tex]
So, [tex]\( A = 51 \)[/tex].
- For [tex]\( T_{11} \)[/tex]:
[tex]\[ T_{11} = d(11^2) + 1 = 2(121) + 1 = 242 + 1 = 243 \][/tex]
So, [tex]\( B = 243 \)[/tex].
- For [tex]\( T_{20} \)[/tex]:
[tex]\[ T_{20} = d(20^2) + 1 = 2(400) + 1 = 800 + 1 = 801 \][/tex]
So, [tex]\( C = 801 \)[/tex].
### 2.3 Determine the 17th term
For [tex]\( T_{17} \)[/tex]:
[tex]\[ T_{17} = d(17^2) + 1 = 2(289) + 1 = 578 + 1 = 579 \][/tex]
So, the 17th term is:
[tex]\[ T_{17} = 579 \][/tex]
### 1.2 Calculate the 36th term
For [tex]\( T_{36} \)[/tex]:
[tex]\[ T_{36} = d(36^2) + 1 = 2(1296) + 1 = 2592 + 1 = 2593 \][/tex]
So, the 36th term is:
[tex]\[ T_{36} = 2593 \][/tex]
### 1.2.3 Which term will be equal to [tex]\(-241\)[/tex]?
We need to solve:
[tex]\[ T_n = -241 \][/tex]
[tex]\[ d \cdot n^2 + 1 = -241 \][/tex]
[tex]\[ 2 \cdot n^2 + 1 = -241 \][/tex]
[tex]\[ 2n^2 + 1 = -241 \][/tex]
[tex]\[ 2n^2 = -242 \][/tex]
[tex]\[ n^2 = -121 \][/tex]
Considering [tex]\( n^2 = -121 \)[/tex] and recognizing that [tex]\( n \)[/tex] must be a real number, there is no real solution for [tex]\( n \)[/tex] because the square of a real number cannot be negative. Therefore, no term in this sequence will be equal to -241 in the context of real numbers.
By following these steps, we've addressed the parts of the problem as a whole.
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