Quadratic nth Term Practice Questions - Corbettmaths - Free Printable

The image you uploaded is the header of a worksheet titled "Quadratic Sequences" from Corbettmaths, which is a resource for mathematics practice. The worksheet provides guidance on how to approach the exam-style questions, and it lists the necessary tools required for solving the problems.

Key Points from the Image:

1. Title: "Quadratic Sequences"
- This indicates that the worksheet focuses on quadratic sequences, which are sequences where the second differences between consecutive terms are constant.

2. Tools Required:
- Pencil
- Pen
- Ruler
- Protractor
- Pair of compasses
- Eraser
- Tracing paper (optional)

3. Guidance:
- Read each question carefully before you begin answering it.
- Don’t spend too long on one question.
- Attempt every question.
- Check your answers seem right.
- Always show your workings.

Explanation of Quadratic Sequences:

A quadratic sequence is a sequence where the difference between consecutive terms forms an arithmetic sequence (i.e., the second differences are constant). For example:

- Consider the sequence: \( 2, 5, 10, 17, 26 \).
- First differences: \( 5-2 = 3 \), \( 10-5 = 5 \), \( 17-10 = 7 \), \( 26-17 = 9 \).
- Second differences: \( 5-3 = 2 \), \( 7-5 = 2 \), \( 9-7 = 2 \) (constant).

The general form of a quadratic sequence can be expressed as:
\[
u_n = an^2 + bn + c
\]
where:
- \( u_n \) is the \( n \)-th term of the sequence,
- \( a \), \( b \), and \( c \) are constants.

Steps to Solve Quadratic Sequence Problems:

1. Identify the first few terms of the sequence.
2. Calculate the first differences between consecutive terms.
3. Calculate the second differences.
- If the second differences are constant, the sequence is quadratic.
4. Use the general form \( u_n = an^2 + bn + c \) to find the constants \( a \), \( b \), and \( c \).
- Substitute known values of \( n \) and \( u_n \) into the equation to form a system of equations.
- Solve the system of equations to find \( a \), \( b \), and \( c \).

Example Problem:

Suppose we are given the sequence: \( 3, 8, 15, 24, 35 \).

#### Step 1: Calculate the first differences.
\[
8 - 3 = 5, \quad 15 - 8 = 7, \quad 24 - 15 = 9, \quad 35 - 24 = 11
\]
First differences: \( 5, 7, 9, 11 \).

#### Step 2: Calculate the second differences.
\[
7 - 5 = 2, \quad 9 - 7 = 2, \quad 11 - 9 = 2
\]
Second differences: \( 2, 2, 2 \) (constant).

#### Step 3: Use the general form \( u_n = an^2 + bn + c \).
We know:
\[
u_1 = 3, \quad u_2 = 8, \quad u_3 = 15
\]

Substitute these values into the general form:
1. For \( n = 1 \):
\[
a(1)^2 + b(1) + c = 3 \implies a + b + c = 3 \quad \text{(Equation 1)}
\]

2. For \( n = 2 \):
\[
a(2)^2 + b(2) + c = 8 \implies 4a + 2b + c = 8 \quad \text{(Equation 2)}
\]

3. For \( n = 3 \):
\[
a(3)^2 + b(3) + c = 15 \implies 9a + 3b + c = 15 \quad \text{(Equation 3)}
\]

#### Step 4: Solve the system of equations.
Subtract Equation 1 from Equation 2:
\[
(4a + 2b + c) - (a + b + c) = 8 - 3 \implies 3a + b = 5 \quad \text{(Equation 4)}
\]

Subtract Equation 2 from Equation 3:
\[
(9a + 3b + c) - (4a + 2b + c) = 15 - 8 \implies 5a + b = 7 \quad \text{(Equation 5)}
\]

Subtract Equation 4 from Equation 5:
\[
(5a + b) - (3a + b) = 7 - 5 \implies 2a = 2 \implies a = 1
\]

Substitute \( a = 1 \) into Equation 4:
\[
3(1) + b = 5 \implies 3 + b = 5 \implies b = 2
\]

Substitute \( a = 1 \) and \( b = 2 \) into Equation 1:
\[
1 + 2 + c = 3 \implies 3 + c = 3 \implies c = 0
\]

Thus, the general term of the sequence is:
\[
u_n = n^2 + 2n
\]

Final Answer:

\[
\boxed{u_n = n^2 + 2n}
\]

This method can be applied to any quadratic sequence problem. Always follow the guidance provided in the worksheet to ensure clarity and accuracy in your solutions.