QUESTION 5 A quadratic pattem has a second term equal to1, a third term equal $$ 10-6 $$ and a fifth term equal to -14 . 5.1 Calculate the second difference of this quadratic pattern. 5.2 Hence, or otherwise, calculate the first term of the pattern.

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We assume the quadratic pattern has terms given by
$$
a_n = An^2 + Bn + C.
$$

**Step 1. Write the equations for the known terms.**

The pattern's given values are:
- The second term: $$ a_2 = 1 $$
- The third term: $$ a_3 = 10 - 6 = 4 $$
- The fifth term: $$ a_5 = -14 $$

Using the general formula:
1. For $$ n = 2 $$:
   $$
   4A + 2B + C = 1
   $$
2. For $$ n = 3 $$:
   $$
   9A + 3B + C = 4
   $$
3. For $$ n = 5 $$:
   $$
   25A + 5B + C = -14
   $$

**Step 2. Eliminate $$ C $$ by subtracting equations.**

Subtract the equation for $$ a_2 $$ from that for $$ a_3 $$:
$$
(9A + 3B + C) - (4A + 2B + C) = 4 - 1
$$
$$
5A + B = 3 \quad \Longrightarrow \quad B = 3 - 5A.
$$

Subtract the equation for $$ a_3 $$ from that for $$ a_5 $$:
$$
(25A + 5B + C) - (9A + 3B + C) = -14 - 4
$$
$$
16A + 2B = -18.
$$
Divide the entire equation by 2:
$$
8A + B = -9.
$$

**Step 3. Solve for $$ A $$ and $$ B $$.**

Substitute $$ B = 3 - 5A $$ into $$ 8A + B = -9 $$:
$$
8A + (3 - 5A) = -9
$$
$$
3A + 3 = -9
$$
$$
3A = -12 \quad \Longrightarrow \quad A = -4.
$$

Now, substitute $$ A = -4 $$ back into $$ B = 3 - 5A $$:
$$
B = 3 - 5(-4) = 3 + 20 = 23.
$$

**Step 4. Determine $$ C $$ using one of the equations.**

Using the equation from $$ a_2 $$:
$$
4A + 2B + C = 1
$$
Substitute $$ A = -4 $$ and $$ B = 23 $$:
$$
4(-4) + 2(23) + C = 1
$$
$$
-16 + 46 + C = 1
$$
$$
30 + C = 1 \quad \Longrightarrow \quad C = 1 - 30 = -29.
$$

Thus, the quadratic pattern is
$$
a_n = -4n^2 + 23n - 29.
$$

**Step 5. Calculate the second difference.**

For any quadratic pattern, the second difference is constant and given by
$$
\text{Second difference} = 2A.
$$
Substitute $$ A = -4 $$:
$$
2A = 2(-4) = -8.
$$

**Step 6. Calculate the first term of the pattern.**

The first term is given by $$ a_1 $$:
$$
a_1 = -4(1)^2 + 23(1) - 29 = -4 + 23 - 29.
$$
$$
a_1 = -10.
$$

**Final Answers:**

5.1 The second difference is $$-8$$.

5.2 The first term of the pattern is $$-10$$.
The second difference is -8, and the first term of the pattern is -10.

QUESTION 5 A quadratic pattem has a second term equal to1, a third term equal \( 10-6 \) and a fifth term equal to -14 . 5.1 Calculate the second difference of this quadratic pattern. 5.2 Hence, or otherwise, calculate the first term of the pattern.

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