https://file.aiquickdraw.com/mathbot/file/1728662333332nw3cjstj.jpg

The given sequence is:  
\[ 2, 4, 8, 14, 22 \]

This is a quadratic sequence, meaning the nth term can be expressed as a quadratic function of the form:
\[
T_n = an^2 + bn + c
\]
where \( T_n \) is the nth term of the sequence, and \( a \), \( b \), and \( c \) are constants to be determined.

### Step 1: First and second differences
Start by finding the first and second differences between consecutive terms.

1st differences:
\[
4 - 2 = 2, \quad 8 - 4 = 4, \quad 14 - 8 = 6, \quad 22 - 14 = 8
\]
So, the first differences are: \( 2, 4, 6, 8 \).

2nd differences (between first differences):
\[
4 - 2 = 2, \quad 6 - 4 = 2, \quad 8 - 6 = 2
\]
The second differences are constant and equal to 2, confirming that the sequence is quadratic.

### Step 2: Using the second difference to find \( a \)
For a quadratic sequence, the second difference is equal to \( 2a \). Therefore:
\[
2a = 2 \quad \Rightarrow \quad a = 1
\]

### Step 3: Setting up the equation
Now that we know \( a = 1 \), the general form of the nth term is:
\[
T_n = n^2 + bn + c
\]

### Step 4: Solving for \( b \) and \( c \)
Substitute known values from the sequence to form equations and solve for \( b \) and \( c \).

For \( n = 1 \), \( T_1 = 2 \):
\[
1^2 + b(1) + c = 2 \quad \Rightarrow \quad 1 + b + c = 2 \quad \Rightarrow \quad b + c = 1 \quad \text{(Equation 1)}
\]

For \( n = 2 \), \( T_2 = 4 \):
\[
2^2 + b(2) + c = 4 \quad \Rightarrow \quad 4 + 2b + c = 4 \quad \Rightarrow \quad 2b + c = 0 \quad \text{(Equation 2)}
\]

### Step 5: Solving the system of equations
From Equation 2: 
\[
2b + c = 0 \quad \Rightarrow \quad c = -2b
\]

Substitute \( c = -2b \) into Equation 1:
\[
b + (-2b) = 1 \quad \Rightarrow \quad -b = 1 \quad \Rightarrow \quad b = -1
\]

Now substitute \( b = -1 \) into \( c = -2b \):
\[
c = -2(-1) = 2
\]

### Step 6: Final expression for \( T_n \)
The nth term of the sequence is:
\[
T_n = n^2 - n + 2
\]

Would you like further clarification or a deeper explanation on any step? 

Here are 5 related questions:
1. How do you verify if a sequence is quadratic?
2. Can a quadratic sequence ever have zero as its second difference?
3. How can you derive a formula for a cubic sequence?
4. What is the significance of first and second differences in sequences?
5. How do changes in \( a \), \( b \), and \( c \) affect the shape of the quadratic sequence?

**Tip:** Always check the second differences of a sequence to confirm it's quadratic. If the second differences are constant, the sequence is quadratic!

Here are the first five terms of a quadratic sequence: 2, 4, 8, 14, 22. Find an expression, in terms of n, for the nth term of the sequence.

Learn how to find the nth term expression of a quadratic sequence with the terms 2, 4, 8, 14, 22. This solution explains key concepts like quadratic sequences, algebraic differences, and includes a step-by-step approach to deriving the formula T_n = n^2 - n + 2. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation