To solve for the constant $k$, we start by using the property of an arithmetic sequence: the difference between consecutive terms is constant.

Given the first three terms of the arithmetic sequence are:

1. $a_1 = k + 2$
2. $a_2 = 2k + 3$
3. $a_3 = 4k - 2$

The common difference $d$ of an arithmetic sequence can be found as:

$d = a_2 - a_1 = a_3 - a_2$

Let's compute each difference:

1. Difference between the second and first terms: $a_2 - a_1 = (2k + 3) - (k + 2) = 2k + 3 - k - 2 = k + 1$

2. Difference between the third and second terms: $a_3 - a_2 = (4k - 2) - (2k + 3) = 4k - 2 - 2k - 3 = 2k - 5$

Since these differences are equal, we set them equal to each other:

$k + 1 = 2k - 5$

Now, solve for $k$:

$k + 1 = 2k - 5$

Subtract $k$ from both sides:

$1 = k - 5$

Add 5 to both sides:

$k = 6$

Conclusion

The value of the constant $k$ is 6.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What are the first five terms of this arithmetic sequence with $k = 6$?
2. How do you find the nth term of an arithmetic sequence?
3. Can you derive a formula for the sum of the first n terms of an arithmetic sequence?
4. What changes in the arithmetic sequence if $k$ were a different value?
5. How can we find the common difference if we only know two non-consecutive terms of the sequence?

Tip: Remember, in arithmetic sequences, knowing any two consecutive terms allows you to find the common difference!