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.

Question

UpStudy AI · Accepted Answer

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.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions