Question

Find k given that (3k + 1), k, and - 3 are the first three terms of an arithmetic sequence. Show your work and please explain what you did in each step.

Answer 1

`k=2`

Explanation:

If the terms form an arithmetic sequence, then there should be a common difference. That is, the difference between the second and first terms should be equal to the difference of the third and second terms.

The difference between the second and first terms is

`k-(3k+1)\\=k-3k-1\\=-2k-1.`

The difference between the third and second terms is

`-3-k.`

Since the differences must be the same for the terms to form an arithmetic sequence, then
`-2k-1=-3-k.` Solving for k results in

`-2k-1=-3-k\\-2k+k=-3+1\\-k=-2\\k=2.`

Hence, the value of k is 2.

Answer 2

• k = 2

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We know that any term of an AP is the average of the terms equidistant from it:

• 
  `t_2 = (t_1+t_3)/2`

or

• 
  `2t_2=t_1+t_3`

Apply this to the given terms and solve for k:

• 2k = 3k + 1 - 3
• 2k = 3k - 2
• 3k - 2k = 2
• k = 2