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.

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.