what must be the value of k so that 5k-3, k+2 and 3k-11 will form an arithmetic sequence?

For the values \(5k - 3\), \(k + 2\), and \(3k - 11\) to form an arithmetic sequence, the difference between consecutive terms must be the same. This means:

\[
(k + 2) - (5k - 3) = (3k - 11) - (k + 2)
\]

Let's solve this step by step.

### Step 1: Calculate the first difference
\[
(k + 2) - (5k - 3) = k + 2 - 5k + 3 = -4k + 5
\]

### Step 2: Calculate the second difference
\[
(3k - 11) - (k + 2) = 3k - 11 - k - 2 = 2k - 13
\]

### Step 3: Set the differences equal to each other
\[
-4k + 5 = 2k - 13
\]

### Step 4: Solve for \(k\)
Add \(4k\) to both sides to get:
\[
5 = 6k - 13
\]

Add \(13\) to both sides:
\[
18 = 6k
\]

Divide by \(6\):
\[
k = 3
\]

Thus, the value of \(k\) must be \(3\).

Would you like to see further details on how this problem was solved, or do you have any other questions?

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Here are 5 related questions to consider:
1. What is the general condition for any three numbers to be in an arithmetic sequence?
2. How would the solution change if the terms were in a geometric sequence instead?
3. Can you form a quadratic equation using the expressions and solve for \(k\)?
4. What happens if the first difference is negative? How would that affect \(k\)?
5. Can the sequence still be arithmetic if the terms are in a different order?

**Tip:** When solving for unknowns in sequences, always equate the common differences for arithmetic sequences or the common ratios for geometric sequences.

Understand how to find the value of k so that the numbers 5k-3, k+2, and 3k-11 form an arithmetic sequence. Step-by-step solution provided.

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