Answer
**QUESTION I**
1. **Calculate the values of \( x \) and \( y \):**
- \( x = 9 \)
- \( y = 17 \)
2. **Determine the \( n^{th} \) term of the sequence:**
- General term: \( T_n = 2n^2 - 2n + 5 \)
3. **Calculate the \( 50^{th} \) term:**
- \( T_{50} = 4905 \)
**QUESTION 2**
1. **Next term of the pattern:**
- \( 4 \)
2. **General term of the pattern:**
- \( T_n = 2n^2 - 2n + 5 \)
3. **Value of the \( 38^{th} \) term:**
- \( T_{38} = 2(38)^2 - 2(38) + 5 = 2(1444) - 76 + 5 = 2888 - 76 + 5 = 2817 \)
Solution
Let's solve the problems step by step.
### QUESTION I
#### Part 1: Calculate the values of \( x \) and \( y \)
We are given a quadratic sequence: \( 5, x, y, 29 \) and the second constant difference is equal to 4.
1. **Understanding the quadratic sequence**:
- The first difference is the difference between consecutive terms.
- The second difference is the difference of the first differences.
2. **Finding the first differences**:
- Let the first differences be \( d_1, d_2, d_3 \):
- \( d_1 = x - 5 \)
- \( d_2 = y - x \)
- \( d_3 = 29 - y \)
3. **Finding the second differences**:
- The second differences are:
- \( d_2 - d_1 = (y - x) - (x - 5) = y - 2x + 5 \)
- \( d_3 - d_2 = (29 - y) - (y - x) = 29 - 2y + x \)
4. **Setting up the equations**:
- Since the second difference is constant and equal to 4, we have:
\[
y - 2x + 5 = 4 \quad (1)
\]
\[
29 - 2y + x = 4 \quad (2)
\]
5. **Solving the equations**:
- From equation (1):
\[
y - 2x + 5 = 4 \implies y - 2x = -1 \implies y = 2x - 1 \quad (3)
\]
- Substitute (3) into equation (2):
\[
29 - 2(2x - 1) + x = 4
\]
\[
29 - 4x + 2 + x = 4
\]
\[
31 - 3x = 4 \implies 3x = 27 \implies x = 9
\]
- Substitute \( x = 9 \) back into (3):
\[
y = 2(9) - 1 = 18 - 1 = 17
\]
Thus, the values of \( x \) and \( y \) are:
\[
x = 9, \quad y = 17
\]
#### Part 2: Determine the \( n^{th} \) term of the quadratic sequence
Given \( x = 9 \) and \( y = 17 \), the sequence is \( 5, 9, 17, 29 \).
1. **Finding the first differences**:
- \( d_1 = 9 - 5 = 4 \)
- \( d_2 = 17 - 9 = 8 \)
- \( d_3 = 29 - 17 = 12 \)
2. **Finding the second differences**:
- \( d_2 - d_1 = 8 - 4 = 4 \)
- \( d_3 - d_2 = 12 - 8 = 4 \)
3. **General term of a quadratic sequence**:
- The general term can be expressed as:
\[
T_n = an^2 + bn + c
\]
- We know the second difference is \( 2a = 4 \implies a = 2 \).
4. **Using known terms to find \( b \) and \( c \)**:
- For \( n = 1 \): \( T_1 = 5 \)
\[
2(1)^2 + b(1) + c = 5 \implies 2 + b + c = 5 \implies b + c = 3 \quad (4)
\]
- For \( n = 2 \): \( T_2 = 9 \)
\[
2(2)^2 + b(2) + c = 9 \implies 8 + 2b + c = 9 \implies 2b + c = 1 \quad (5)
\]
5. **Solving equations (4) and (5)**:
- From (4): \( c = 3 - b \)
- Substitute into (5):
\[
2b + (3 - b) = 1 \implies 2b + 3 - b = 1 \implies b + 3 = 1 \implies b = -2
\]
- Substitute \( b = -2 \) back into (4):
\[
-2 + c = 3 \implies c = 5
\]
Thus, the general term is:
\[
T_n = 2n^2 - 2n + 5
\]
#### Part 3: Calculate the \( 50^{th} \) term of the sequence
To find \( T_{50} \):
\[
T_{50} = 2(50)^2 - 2(50) + 5
\]
Calculating:
\[
T_{50} = 2(2500) - 100 + 5 = 5000 - 100 + 5 = 4905
\]
### QUESTION 2
#### Part 1: Write down the next term of the pattern
The given sequence is: \( -16, -16, -12, -4 \).
1. **Finding the first differences**:
- \( -16 - (-16) = 0 \)
- \( -12 - (-16) = 4 \)
- \( -4 - (-12) = 8 \)
2. **Finding the second differences**:
- \( 4 - 0 = 4 \)
- \( 8 - 4 = 4 \)
Since the second difference is constant, the next term can be found by adding the last first difference (which is 8) to the last term (-4):
\[
-4 + 8 = 4
\]
Thus, the next term is:
\[
4
\]
#### Part 2: Determine the general term of the pattern
1. **Using the known terms**:
- The terms are \( -16, -16, -12, -4, 4 \).
- We can express the general term as:
\[
T_n = an^2 + bn + c
\]
2. **Setting up equations**:
- For \( n = 1 \): \( T_1 = -16 \)
\[
a(1)^2 + b(1) + c = -16 \quad (6)
\]
- For \( n = 2 \): \( T_2 = -16 \
Reviewed and approved by the UpStudy tutoring team
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