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Let's break down each part of this question and solve it step by step:

### 2.14.1 First difference between \( T_{37} \) and \( T_{38} \)
The first differences provided are \( 2, 6, 10, \ldots \), and these suggest that the differences are increasing by 4 each time, meaning the quadratic pattern follows a certain second difference pattern.

The general quadratic sequence has the form:

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

The first differences between consecutive terms follow a linear pattern:

\[
\Delta T_n = T_{n+1} - T_n = 2a(n) + b
\]

Given that the first differences are \( 2, 6, 10, \ldots \), it suggests that the differences follow an arithmetic sequence with a common difference of 4.

To find the first difference between \( T_{37} \) and \( T_{38} \), we'll need the general term of the first difference, i.e.,

\[
\Delta T_n = 2a(n) + b
\]

Since the first difference follows a linear pattern starting at 2 and increasing by 4, this implies:

\[
\Delta T_n = 4n - 2
\]

Thus, the first difference between \( T_{37} \) and \( T_{38} \) is:

\[
\Delta T_{37} = 4(37) - 2 = 148 - 2 = 146
\]
\[
\Delta T_{38} = 4(38) - 2 = 152 - 2 = 150
\]

So, the first difference between \( T_{37} \) and \( T_{38} \) is **150 - 146 = 4**.

### 2.14.2 Proof of the general term given the 19th term is 651
We are given that the general term of the quadratic pattern is \( T_n = 2n^2 - 4n + 5 \), and the 19th term \( T_{19} = 651 \).

Let's substitute \( n = 19 \) into the given expression:

\[
T_{19} = 2(19)^2 - 4(19) + 5
\]
\[
T_{19} = 2(361) - 76 + 5 = 722 - 76 + 5 = 651
\]

Hence, the given general term \( T_n = 2n^2 - 4n + 5 \) is correct.

### 2.14.3 Prove that all terms of the quadratic pattern will be odd
The general term is \( T_n = 2n^2 - 4n + 5 \).

Let's analyze the expression modulo 2 (since we're interested in whether the terms are odd).

\[
T_n = 2n^2 - 4n + 5
\]

For any integer \( n \):
- \( 2n^2 \) is always even.
- \( -4n \) is always even.
- \( +5 \) is odd.

So, \( T_n \) simplifies to:

\[
T_n = \text{even} - \text{even} + \text{odd} = \text{odd}
\]

Thus, all terms of the quadratic pattern will indeed be odd.

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Would you like more details or explanations on any of the steps?

### Related Questions:
1. What is the second difference for a quadratic sequence, and why is it constant?
2. How do you derive the general term of a quadratic sequence from the first and second differences?
3. Can you explain how to verify a general term by substituting values of \( n \)?
4. How do arithmetic progressions relate to first differences in sequences?
5. Can you explore the relationship between even and odd terms in quadratic patterns?

### Tip:
When working with quadratic sequences, always check the first and second differences to identify patterns and help determine the general formula.

2.14 2; 6; 10 ... are the first differences of a quadratic pattern. 
2.14.1 Determine the first difference between T_37 and T_38 of the quadratic pattern. 
2.14.2 If the 19th term of a quadratic pattern is 651, prove that the general term will be given by T_n = 2n^2 - 4n + 5. 
2.14.3 Prove that all terms of the quadratic pattern will be odd.

Understand how to solve problems involving quadratic patterns, including finding the first difference between terms, proving a general formula, and determining properties of the sequence. Suitable for high school students in Grades 9-12.

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