Worksheet for finding the nth term of quadratic sequences with 13 example problems.

Let's solve Problem 1 from the list:
Sequence: $ 2, 8, 18, 32, 50, \ldots $

We are to find the nth term of this quadratic sequence.

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🔍 Step 1: Check if it's a quadratic sequence

A quadratic sequence has a constant second difference.

Let’s compute the first differences and then the second differences.

Given sequence:
$$
a_1 = 2,\quad a_2 = 8,\quad a_3 = 18,\quad a_4 = 32,\quad a_5 = 50
$$

#### First differences (between consecutive terms):
$$
8 - 2 = 6 \\
18 - 8 = 10 \\
32 - 18 = 14 \\
50 - 32 = 18
$$
So first differences: $ 6, 10, 14, 18 $

#### Second differences:
$$
10 - 6 = 4 \\
14 - 10 = 4 \\
18 - 14 = 4
$$
Second differences are constant: $ 4 $

✔ So this is a quadratic sequence.

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🧮 Step 2: General form of nth term

For a quadratic sequence, the nth term is of the form:
$$
a_n = an^2 + bn + c
$$
We need to find constants $ a $, $ b $, and $ c $.

We’ll use the first three terms to set up equations.

Let’s plug in $ n = 1, 2, 3 $:

#### For $ n = 1 $: $ a(1)^2 + b(1) + c = 2 $
$$
a + b + c = 2 \quad \text{(Equation 1)}
$$

#### For $ n = 2 $: $ a(4) + b(2) + c = 8 $
$$
4a + 2b + c = 8 \quad \text{(Equation 2)}
$$

#### For $ n = 3 $: $ a(9) + b(3) + c = 18 $
$$
9a + 3b + c = 18 \quad \text{(Equation 3)}
$$

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✏️ Step 3: Solve the system of equations

Subtract Equation 1 from Equation 2:
$$
(4a + 2b + c) - (a + b + c) = 8 - 2 \\
3a + b = 6 \quad \text{(Equation 4)}
$$

Subtract Equation 2 from Equation 3:
$$
(9a + 3b + c) - (4a + 2b + c) = 18 - 8 \\
5a + b = 10 \quad \text{(Equation 5)}
$$

Now subtract Equation 4 from Equation 5:
$$
(5a + b) - (3a + b) = 10 - 6 \\
2a = 4 \Rightarrow a = 2
$$

Plug $ a = 2 $ into Equation 4:
$$
3(2) + b = 6 \Rightarrow 6 + b = 6 \Rightarrow b = 0
$$

Now plug $ a = 2 $, $ b = 0 $ into Equation 1:
$$
2 + 0 + c = 2 \Rightarrow c = 0
$$

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✔ Final Answer:

$$
a_n = 2n^2
$$

Let’s verify with the given terms:

- $ n=1 $: $ 2(1)^2 = 2 $ ✔
- $ n=2 $: $ 2(4) = 8 $ ✔
- $ n=3 $: $ 2(9) = 18 $ ✔
- $ n=4 $: $ 2(16) = 32 $ ✔
- $ n=5 $: $ 2(25) = 50 $ ✔

Perfect!

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🟩 Answer for Problem 1:

$$
\boxed{a_n = 2n^2}
$$

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Would you like me to solve the other problems as well? I can go through them one by one.