Find the general term of 2, 7, 16, 29, 46, ...

The sequence given is:

\[ 2, 7, 16, 29, 46, \dots \]

To find the general term, we observe how the terms change:

- The difference between the second and first term: \( 7 - 2 = 5 \)
- The difference between the third and second term: \( 16 - 7 = 9 \)
- The difference between the fourth and third term: \( 29 - 16 = 13 \)
- The difference between the fifth and fourth term: \( 46 - 29 = 17 \)

We can see that the differences are increasing by 4 each time, suggesting that this is a quadratic sequence.

### Step 1: Assume a quadratic form for the general term
The general term of a quadratic sequence is given by:

\[
T_n = an^2 + bn + c
\]

### Step 2: Use known values to form equations
We know the first few terms of the sequence, so substitute \(n = 1, 2, 3\) into the equation:

1. \( T_1 = 2 \): 
\[
a(1)^2 + b(1) + c = 2 \quad \Rightarrow \quad a + b + c = 2
\]

2. \( T_2 = 7 \):
\[
a(2)^2 + b(2) + c = 7 \quad \Rightarrow \quad 4a + 2b + c = 7
\]

3. \( T_3 = 16 \):
\[
a(3)^2 + b(3) + c = 16 \quad \Rightarrow \quad 9a + 3b + c = 16
\]

### Step 3: Solve the system of equations
Now we have the following system of equations:

1. \( a + b + c = 2 \)
2. \( 4a + 2b + c = 7 \)
3. \( 9a + 3b + c = 16 \)

Subtract the first equation from the second to eliminate \(c\):

\[
(4a + 2b + c) - (a + b + c) = 7 - 2 \quad \Rightarrow \quad 3a + b = 5 \quad \text{(Equation 4)}
\]

Subtract the second equation from the third:

\[
(9a + 3b + c) - (4a + 2b + c) = 16 - 7 \quad \Rightarrow \quad 5a + b = 9 \quad \text{(Equation 5)}
\]

### Step 4: Solve for \(a\) and \(b\)
Subtract Equation 4 from Equation 5:

\[
(5a + b) - (3a + b) = 9 - 5 \quad \Rightarrow \quad 2a = 4 \quad \Rightarrow \quad a = 2
\]

Now substitute \(a = 2\) into Equation 4:

\[
3(2) + b = 5 \quad \Rightarrow \quad 6 + b = 5 \quad \Rightarrow \quad b = -1
\]

### Step 5: Solve for \(c\)
Substitute \(a = 2\) and \(b = -1\) into the first equation:

\[
2 - 1 + c = 2 \quad \Rightarrow \quad c = 1
\]

### Step 6: Write the general term
Thus, the general term for the sequence is:

\[
T_n = 2n^2 - n + 1
\]

### Verification:
- For \(n = 1\): \(T_1 = 2(1)^2 - 1 + 1 = 2\)
- For \(n = 2\): \(T_2 = 2(2)^2 - 2 + 1 = 7\)
- For \(n = 3\): \(T_3 = 2(3)^2 - 3 + 1 = 16\)
- For \(n = 4\): \(T_4 = 2(4)^2 - 4 + 1 = 29\)
- For \(n = 5\): \(T_5 = 2(5)^2 - 5 + 1 = 46\)

The general term is correct.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do we determine the degree of a sequence?
2. Can we apply this method to find the general term of cubic sequences?
3. How do differences between consecutive terms indicate the type of sequence?
4. What are other methods for finding general terms of sequences?
5. How can we solve quadratic equations using matrices?

**Tip**: Always check the differences between consecutive terms to identify the degree of the sequence!

Discover how to find the general term for the quadratic sequence 2, 7, 16, 29, 46 with a clear, step-by-step explanation. Learn key concepts like quadratic sequences and solving systems of equations. This solution is ideal for students in Grades 9-11.

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