What is the value of k for which 5k + 2, 4k - 1 and k + 2 are in arithmetic progression?a)2b)3c)4d)6Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Here 4k - 1 - (5k + 2) = (k + 2) - (4k - 1)
⇒ 4k - 1 - 5k - 2 = k + 2 - 4k + 1
⇒ -k - 3 = -3k + 3
⇒ 2k = 6 ⇒ k = 3

Answer

To find the value of k for which the given numbers are in arithmetic progression, we need to use the formula for the nth term of an arithmetic progression.

The formula for the nth term of an arithmetic progression is given by:

an = a + (n-1)d

Where:
an = nth term
a = first term
n = term number
d = common difference

In this case, we have three terms: 5k + 2, 4k - 1, and k + 2. We can use these terms to set up a system of equations and solve for k.

Let's set up the equations:

First term: a = 5k + 2
Second term: a + d = 4k - 1
Third term: a + 2d = k + 2

Now, we can solve this system of equations to find the value of k.

Substituting the first equation into the second equation, we get:

5k + 2 + d = 4k - 1

Rearranging the equation, we have:

d = 4k - 1 - 5k - 2
d = -k - 3

Substituting the first equation into the third equation, we get:

5k + 2 + 2d = k + 2

Rearranging the equation, we have:

2d = k + 2 - 5k - 2
2d = -4k

Now we have two equations involving d:

d = -k - 3
2d = -4k

We can solve these equations simultaneously by substituting the value of d from the first equation into the second equation:

2(-k - 3) = -4k
-2k - 6 = -4k

Simplifying the equation, we get:

-2k + 4k = 6
2k = 6
k = 3

Therefore, the value of k for which the given numbers are in arithmetic progression is 3, which corresponds to option B.

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