Introduction:
To find the value of 'k' in an arithmetic progression (AP) with the terms 2/3, k, and 5k/8, we can use the formula for the nth term of an AP. Since we have three terms, we can compare the differences between consecutive terms to determine the common difference.

Step 1: Identify the terms:
The given terms in the AP are:
Term 1 (a): 2/3
Term 2 (a + d): k
Term 3 (a + 2d): 5k/8

Step 2: Calculate the common difference:
To calculate the common difference (d), we can use the formula:
Common difference (d) = Term 2 - Term 1

So, d = k - 2/3

Step 3: Check if the terms are in AP:
To verify if the terms are in AP, we need to check if the differences between consecutive terms are constant. We can compare the differences between Term 2 and Term 1, and Term 3 and Term 2.

Difference between Term 2 and Term 1:
k - 2/3

Difference between Term 3 and Term 2:
(5k/8) - k = 5k/8 - 8k/8 = -3k/8

If the differences are equal, then the terms are in AP. Therefore, we have:
k - 2/3 = -3k/8

Step 4: Solve for k:
To find the value of 'k', we can solve the equation obtained in Step 3.

Multiplying both sides of the equation by 24 (to eliminate fractions):
24(k - 2/3) = 24(-3k/8)

Simplifying the equation:
8k - 16 = -9k

Adding 9k to both sides:
8k + 9k - 16 = 0
17k - 16 = 0

Adding 16 to both sides:
17k = 16
k = 16/17

Conclusion:
The value of 'k' that satisfies the given condition and makes the terms 2/3, k, and 5k/8 form an arithmetic progression is k = 16/17.