The given arithmetic series is: $2 + 8 + 14 + \ldots$

a) Finding the value of $k$ where the $k$-th term is 332

In an arithmetic series, the $n$-th term $a_n$ is given by:

$a_n = a_1 + (n-1) \cdot d$

Where:

• $a_1$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

For the given series:

• $a_1 = 2$
• The common difference $d = 8 - 2 = 6$

So, the general term formula becomes:

$a_n = 2 + (n-1) \cdot 6$

We want to find $k$ such that $a_k = 332$:

$332 = 2 + (k-1) \cdot 6$

Simplifying the equation:

$332 - 2 = (k-1) \cdot 6$ $330 = (k-1) \cdot 6$ $k - 1 = \frac{330}{6}$ $k - 1 = 55$ $k = 56$

So, the $56$-th term of the arithmetic series is 332.

b) Forming the quadratic equation

We are asked to form a quadratic equation where the sum of the roots is equal to the $10$-th term of the series.

From the formula for the $n$-th term: $a_{10} = 2 + (10-1) \cdot 6 = 2 + 9 \cdot 6 = 2 + 54 = 56$

So, the sum of the roots of the quadratic equation is 56.

For a quadratic equation $ay^2 + by + c = 0$, the sum of the roots is given by:

$-\frac{b}{a} = 56$

This implies:

$b = -56a$

Thus, the quadratic equation will have the form:

$ay^2 - 56ay + c = 0$

If you have more details for $c$, the equation can be fully determined. Otherwise, this is the general form.

Would you like more details on any part of the solution?

Here are some follow-up questions to consider:

1. How do you derive the general term of an arithmetic sequence?
2. What is the relationship between the common difference and the sequence terms?
3. How do you find the sum of the first $n$ terms of an arithmetic series?
4. What is the significance of the discriminant in a quadratic equation?
5. How can you determine the product of the roots from the quadratic equation?

Tip: In quadratic equations, the product of the roots is given by $\frac{c}{a}$, which helps when factoring or solving.