Understanding the Arithmetic Sequence and Solving for k

Recently, a problem involving the value of k that makes certain terms form an arithmetic sequence has gained attention. Specifically, it was asked whether the values 22k 1, 22k, and 32k 1/2 form an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is a constant.

Determining the Value of k

To solve for k, we first set up the terms and their differences. The terms are:

term_1 22k 1 term_2 22k term_3 32k 1/2

We know that the common difference, d, in an arithmetic sequence must be the same between every pair of consecutive terms:

d term_2 — term_1 term_3 — term_2

Step 1: Calculate the Common Difference

The common difference is given by:

d 22k - 22k 1

Note that 22k 1 2 22k, thus:

d 22k - 2 2k -2 2k

Step 2: Equate the Differences

Next, we equate the difference between the first and second terms with the difference between the second and third terms:

d 32k 1/2 - 22k

We already have d -2 2k, so:

-2 2k 32k 1/2 - 22k

Step 3: Solve for k

Now, solve for k:

-2 2k 32k 1/2 - 22k

-2 2k 22k 32k 1/2

-2 2k 22k - 32k 1/2

22k(-2 1 - 3) 1/2

22k(-4) 1/2

22k -1/8

2k -3/2

k -3/4

Conclusion

By analyzing and solving the equations, we find that the value of k is -3/4. The terms corresponding to this k are:

term_1 22(-3/4) 1 2-1/2 1/√2 ≈ -3 term_2 22(-3/4) 2-3/2 1/23/2 1/2√2 ≈ -5 term_3 32(-3/4) 1/2 3-3/2 1/2 ≈ -7

Additional Scenarios for Different Sequences

Additionally, consider the following scenarios based on different configurations of the sequence terms:

Scenario 1: k -2.5

In this case, the terms form an arithmetic sequence as follows:

term_1 2-2.5 1 -3 term_2 2-2.5 -5 term_3 3-2.5 1/2 -7

The common difference for this sequence is -2, and the sequence is: -3, -5, -7, -9, -11, -13, -15, …

Scenario 2: Positive k 3.5

For a positive value of k, the sequence becomes:

term_1 23.5 1 7 term_2 23.5 9 term_3 33.5 1/2 11

The sequence in this case is: 7, 9, 11, … and the common difference is 2.

Scenario 3: Central Term as 32k 1/2

Finally, if the central term is 32k 1/2, the value of k can be 0.5, resulting in:

term_1 2(2(0.5) 1) 3 term_2 2(2(0.5)) 2 term_3 3(2(0.5)) 1/2 1

The sequence here would be: 3, 2, 1, … with a common difference of -1.