EXERCISE I (a) Determine the general term of each of the following quadratic number patterns: (1) $$ 8 ; 13 ; 20 ; 29 ; \ldots $$ (3) $$ \quad 0 ; \frac{5}{2} ; 6 ; \frac{21}{2} ; \ldots $$ (b) Consider the quadratic number pattern $$ -1 ; 1 ; 7 ; 17 ; \ldots $$ (1) Calculate the value of the $$ 50^{ ext {th }} $$ term in this pattern. (2) Determine $$ n $$ if $$ T_{n}=6961 $$. (4) $$ -1 ;-4 \frac{1}{2} ;-10 ;-17 \frac{1}{2} ; \ldots $$ (c) Consider the quadratic number pattern $$ 0 ;-2 ;-5 ;-9 ; \ldots $$ (1) Calculate the value of $$ T_{25} $$ : (2) Which term in this pattern is equal to -1034 ? (d) Solve for $$ x $$ in the following quadratic number patterns: (1) $$ \quad 2 ; 8 ; x ; 26 ; \ldots $$ (2) $$ \quad 6 ; 3 x-1 ; x ;-3 ; \ldots $$ (e) If $$ 3 ; x ; 11 ; 21 ; y ; \ldots $$ is a quadratic number pattern, determine the values of $$ x $$ and $$ y $$. (f) The general term of a quadratic number pattern is given by $$ T_{n}=-(n-50)^{2}+20 $$. (1) What is the constant second difference of the number pattern? (2) Which term has the highest value and what is the value of this term? (2) $$ \quad 2 $$

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EXERCISE I (a) Determine the general term of each of the following quadratic number patterns: (1) $$ 8 ; 13 ; 20 ; 29 ; \ldots $$ (3) $$ \quad 0 ; \frac{5}{2} ; 6 ; \frac{21}{2} ; \ldots $$ (b) Consider the quadratic number pattern $$ -1 ; 1 ; 7 ; 17 ; \ldots $$ (1) Calculate the value of the $$ 50^{	ext {th }} $$ term in this pattern. (2) Determine $$ n $$ if $$ T_{n}=6961 $$. (4) $$ -1 ;-4 \frac{1}{2} ;-10 ;-17 \frac{1}{2} ; \ldots $$ (c) Consider the quadratic number pattern $$ 0 ;-2 ;-5 ;-9 ; \ldots $$ (1) Calculate the value of $$ T_{25} $$ : (2) Which term in this pattern is equal to -1034 ? (d) Solve for $$ x $$ in the following quadratic number patterns: (1) $$ \quad 2 ; 8 ; x ; 26 ; \ldots $$ (2) $$ \quad 6 ; 3 x-1 ; x ;-3 ; \ldots $$ (e) If $$ 3 ; x ; 11 ; 21 ; y ; \ldots $$ is a quadratic number pattern, determine the values of $$ x $$ and $$ y $$. (f) The general term of a quadratic number pattern is given by $$ T_{n}=-(n-50)^{2}+20 $$. (1) What is the constant second difference of the number pattern? (2) Which term has the highest value and what is the value of this term? (2) $$ \quad 2 $$

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Let's solve the exercise step by step.

### (a) Determine the general term of each of the following quadratic number patterns:

#### (1) $$ 8 ; 13 ; 20 ; 29 ; \ldots $$

To find the general term, we first calculate the first and second differences:

- First differences: 
  - $$ 13 - 8 = 5 $$
  - $$ 20 - 13 = 7 $$
  - $$ 29 - 20 = 9 $$

- Second differences:
  - $$ 7 - 5 = 2 $$
  - $$ 9 - 7 = 2 $$

Since the second difference is constant, the sequence is quadratic. The general term can be expressed as:
$$
T_n = an^2 + bn + c
$$

Using the first three terms to set up equations:
1. $$ T_1 = a(1)^2 + b(1) + c = 8 $$
2. $$ T_2 = a(2)^2 + b(2) + c = 13 $$
3. $$ T_3 = a(3)^2 + b(3) + c = 20 $$

This gives us the system of equations:
1. $$ a + b + c = 8 $$
2. $$ 4a + 2b + c = 13 $$
3. $$ 9a + 3b + c = 20 $$

Now, we can solve this system of equations.

#### (3) $$ 0 ; \frac{5}{2} ; 6 ; \frac{21}{2} ; \ldots $$

Calculating the first and second differences:

- First differences:
  - $$ \frac{5}{2} - 0 = \frac{5}{2} $$
  - $$ 6 - \frac{5}{2} = \frac{7}{2} $$
  - $$ \frac{21}{2} - 6 = \frac{9}{2} $$

- Second differences:
  - $$ \frac{7}{2} - \frac{5}{2} = 1 $$
  - $$ \frac{9}{2} - \frac{7}{2} = 1 $$

Since the second difference is constant, we can express the general term as:
$$
T_n = an^2 + bn + c
$$

Using the first three terms:
1. $$ T_1 = 0 $$
2. $$ T_2 = \frac{5}{2} $$
3. $$ T_3 = 6 $$

This gives us the system of equations:
1. $$ a + b + c = 0 $$
2. $$ 4a + 2b + c = \frac{5}{2} $$
3. $$ 9a + 3b + c = 6 $$

Now, we can solve this system of equations.

### (b) Consider the quadratic number pattern $$ -1 ; 1 ; 7 ; 17 ; \ldots $$

#### (1) Calculate the value of the $$ 50^{\text{th}} $$ term in this pattern.

Calculating the first and second differences:

- First differences:
  - $$ 1 - (-1) = 2 $$
  - $$ 7 - 1 = 6 $$
  - $$ 17 - 7 = 10 $$

- Second differences:
  - $$ 6 - 2 = 4 $$
  - $$ 10 - 6 = 4 $$

The second difference is constant, so we can express the general term as:
$$
T_n = an^2 + bn + c
$$

Using the first three terms:
1. $$ T_1 = -1 $$
2. $$ T_2 = 1 $$
3. $$ T_3 = 7 $$

This gives us the system of equations:
1. $$ a + b + c = -1 $$
2. $$ 4a + 2b + c = 1 $$
3. $$ 9a + 3b + c = 7 $$

Now, we can solve this system of equations to find $$ a, b, c $$ and then calculate $$ T_{50} $$.

#### (2) Determine $$ n $$ if $$ T_{n}=6961 $$.

Once we have the general term, we can set $$ T_n = 6961 $$ and solve for $$ n $$.

#### (4) $$ -1 ; -4 \frac{1}{2} ; -10 ; -17 \frac{1}{2} ; \ldots $$

Calculating the first and second differences:

- First differences:
  - $$ -4.5 - (-1) = -3.5 $$
  - $$ -10 - (-4.5) = -5.5 $$
  - $$ -17.5 - (-10) = -7.5 $$

- Second differences:
  - $$ -5.5 - (-3.5) = -2 $$
  - $$ -7.5 - (-5.5) = -2 $$

The second difference is constant, so we can express the general term as:
$$
T_n = an^2 + bn + c
$$

Using the first three terms:
1. $$ T_1 = -1 $$
2. $$ T_2 = -4.5 $$
3. $$ T_3 = -10 $$

This gives us the system of equations:
1. $$ a + b + c = -1 $$
2. $$ 4a + 2b + c = -4.5 $$
3. $$ 9a + 3b + c = -10 $$

Now, we can solve this system of equations.

### (c) Consider the quadratic number pattern $$ 0 ; -2 ; -5 ; -9 ; \ldots $$

#### (1) Calculate the value of $$ T_{25} $$.

Calculating the first and second differences:

- First differences:
  - $$ -2 - 0 = -2 $$
  - $$ -5 - (-2) = -3 $$
  - $$ -9 - (-5) = -4 $$

- Second differences:
  - $$ -3 - (-2) = -1 $$
  - $$ -4 - (-3) = -1 $$

The second difference is constant, so we can express the general term as:
$$
T_n = an^2 + bn + c
$$

Using the first three terms:
1. $$ T_1 = 0 $$
2. $$ T_2 = -2 $$
3. $$ T_3 = -5 $$

This gives us the system of equations:
1. $$ a + b + c = 0 $$
2. $$ 4a + 2b + c = -2 $$
3. $$ 9a + 3b + c = -5 $$

Now, we can solve this system of equations to find
(a) The general terms are: 1. $$ T_n = n^2 + 2n - 1 $$ 3. $$ T_n = \frac{5}{2}n^2 - \frac{3}{2}n $$ (b) 1. $$ T_{50} = 2451 $$ 2. $$ n = 80 $$ 4. $$ T_n = -n^2 + 10n - 1 $$ (c) 1. $$ T_{25} = -295 $$ 2. The term equal to -1034 is the 51st term. (d) 1. $$ x = 14 $$ 2. $$ x = 2 $$ (e) $$ x = 4 $$, $$ y = 25 $$ (f) 1. The constant second difference is 2. 2. The term with the highest value is the 50th term, which is 20. 3. $$ T_n = -(n-50)^2 + 20 $$

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