Question
Activity 4 4.1 Consider the linear sequence: \( 5 ; 8 ; 11 ; b: 17: \ldots \) 4.1.1 Write down the value of \( b \). 4.1.2 Determine the \( n^{t h} \) term of the sequence. 4.1.3 Calculate the value of the \( 15^{\oplus} \) term of the sequence. 4.1.4 Which term of the sequence is equal to 83 ? Activity 5 5.1 The first term of a linear number pattern is 92 and the constant difference is -4 . 5.1.1 Write down the values of the second and third terms of the pattern. 5.12 Determine an expression for the \( n^{\text {th }} \) term of the number patterm. 5.13 Determine the value of the eighteenth term. S.1.4 If \( T_{p}+T_{q}=0 \), determine the value of \( (p+q) \). TUTORLAL 2 Activity 1 1.1 Given the finite linear number pattern: \( 12 ; 17 ; 22 ; 22 ; \ldots ; 172 \). 1.1.1 Deternine the formula \( n^{\text {th }} \) term of the pattern. 1.1.2 Calculate the value of \( T_{12} \). 1.1.3 Deternine the number of terms in the pattern. 12 Given the first four terms of a linear pattern: \( 3: x: y: 30 \)
Ask by Gough Mejia. in South Africa
Feb 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
**Activity 4**
1. **Value of \( b \):** 14
2. **\( n^{th} \) term formula:** \( T_n = 3n + 2 \)
3. **15th term:** 47
4. **Term equal to 83:** 27th term
**Activity 5**
1. **Second term:** 88
2. **Third term:** 84
3. **\( n^{th} \) term formula:** \( T_n = 96 - 4n \)
4. **18th term:** 24
5. **If \( T_p + T_q = 0 \), then \( p + q = 48 \)**
**Tutorial 2**
1. **\( n^{th} \) term formula:** \( T_n = 5n + 7 \)
2. **12th term:** 67
3. **Number of terms:** 33
**Given the first four terms of a linear pattern: 3, x, y, 30**
- **x:** 12
- **y:** 21
Solution
Let's solve the problems step by step.
### Activity 4
#### 4.1 Consider the linear sequence: \( 5 ; 8 ; 11 ; b ; 17 ; \ldots \)
**4.1.1 Write down the value of \( b \).**
The sequence is linear, meaning it has a constant difference.
- The difference between the first two terms: \( 8 - 5 = 3 \)
- The difference between the second and third terms: \( 11 - 8 = 3 \)
- Therefore, the common difference \( d = 3 \).
To find \( b \):
\[
b = 11 + 3 = 14
\]
**4.1.2 Determine the \( n^{th} \) term of the sequence.**
The \( n^{th} \) term of a linear sequence can be expressed as:
\[
T_n = a + (n-1)d
\]
where \( a \) is the first term and \( d \) is the common difference.
Here, \( a = 5 \) and \( d = 3 \):
\[
T_n = 5 + (n-1) \cdot 3 = 5 + 3n - 3 = 3n + 2
\]
**4.1.3 Calculate the value of the \( 15^{th} \) term of the sequence.**
Using the formula for the \( n^{th} \) term:
\[
T_{15} = 3(15) + 2 = 45 + 2 = 47
\]
**4.1.4 Which term of the sequence is equal to 83?**
Set the \( n^{th} \) term equal to 83:
\[
3n + 2 = 83
\]
\[
3n = 81 \implies n = 27
\]
### Activity 5
#### 5.1 The first term of a linear number pattern is 92 and the constant difference is -4.
**5.1.1 Write down the values of the second and third terms of the pattern.**
- Second term: \( 92 - 4 = 88 \)
- Third term: \( 88 - 4 = 84 \)
**5.1.2 Determine an expression for the \( n^{th} \) term of the number pattern.**
Using the formula for the \( n^{th} \) term:
\[
T_n = a + (n-1)d
\]
where \( a = 92 \) and \( d = -4 \):
\[
T_n = 92 + (n-1)(-4) = 92 - 4n + 4 = 96 - 4n
\]
**5.1.3 Determine the value of the eighteenth term.**
Using the formula:
\[
T_{18} = 96 - 4(18) = 96 - 72 = 24
\]
**5.1.4 If \( T_{p} + T_{q} = 0 \), determine the value of \( (p + q) \).**
Using the expression for \( T_n \):
\[
T_p + T_q = (96 - 4p) + (96 - 4q) = 0
\]
\[
192 - 4(p + q) = 0 \implies 4(p + q) = 192 \implies p + q = 48
\]
### TUTORIAL 2
#### Activity 1
**1.1 Given the finite linear number pattern: \( 12 ; 17 ; 22 ; \ldots ; 172 \).**
**1.1.1 Determine the formula \( n^{th} \) term of the pattern.**
The common difference \( d \) is:
\[
17 - 12 = 5
\]
Using the formula for the \( n^{th} \) term:
\[
T_n = a + (n-1)d
\]
where \( a = 12 \) and \( d = 5 \):
\[
T_n = 12 + (n-1) \cdot 5 = 12 + 5n - 5 = 5n + 7
\]
**1.1.2 Calculate the value of \( T_{12} \).**
Using the formula:
\[
T_{12} = 5(12) + 7 = 60 + 7 = 67
\]
**1.1.3 Determine the number of terms in the pattern.**
Set \( T_n = 172 \):
\[
5n + 7 = 172
\]
\[
5n = 165 \implies n = 33
\]
### 12 Given the first four terms of a linear pattern: \( 3 ; x ; y ; 30 \)
To find \( x \) and \( y \), we need to determine the common difference.
Let \( d \) be the common difference:
- From \( 3 \) to \( x \): \( x = 3 + d \)
- From \( x \) to \( y \): \( y = x + d = 3 + 2d \)
- From \( y \) to \( 30 \): \( 30 = y + d = 3 + 3d \)
Setting up the equation:
\[
3 + 3d = 30 \implies 3d = 27 \implies d = 9
\]
Now substituting back to find \( x \) and \( y \):
\[
x = 3 + 9 = 12
\]
\[
y = 3 + 2(9) = 21
\]
Thus, the values are:
- \( x = 12 \)
- \( y = 21 \)
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
In the linear sequence \(5; 8; 11; b; 17;\ldots\), we can see that the common difference is 3. Therefore, to find \(b\), we simply add 3 to the previous term: \(11 + 3 = 14\). So, \(b = 14\). For the \(n^{th}\) term of this sequence, we can express it as \(T_n = 5 + (n - 1) \times 3\), simplifying to \(T_n = 3n + 2\). Now, onto the second activity! The first term is 92 and the common difference is -4. To find the second term, we calculate \(92 - 4 = 88\) and the third term would be \(88 - 4 = 84\). For the \(n^{th}\) term, we use the formula \(T_n = 92 - 4(n - 1)\), which simplifies to \(T_n = 96 - 4n\). To find the eighteenth term, we plug in \(n = 18\): \(T_{18} = 96 - 4 \times 18 = 96 - 72 = 24\). If \(T_p + T_q = 0\), you can find \(p + q\) by setting \(96 - 4p + 96 - 4q = 0\), leading to \(p + q = 48\). Saddle up for some fun math exploration!
