Given that k,3k and k+1 are three consecutive terms of an A.P,find the numerical value of k.Then find the first term if 3k is the sixth term of the progression.

Answer:k=1over4
a=-7over4
I got the k but the a i can't get

first condition:

3k - k = k+1 - 3k
4k = 1
k = 1/4, you had that

so we have
*, *, *, *, k , 3k, k+1

a+5d = 3k = 3/4
a + 4d = k = 1/4
subtract them
d = 1/2

in a+4d = 1/4
a + 2 = 1/4
a = 1/4-2 = -7/4

Thank you..

To find the value of k, we know that k, 3k, and k+1 are three consecutive terms of an arithmetic progression (AP). In an AP, the common difference (d) is the difference between any two consecutive terms.

So, the difference between the second term (3k) and the first term (k) is equal to the difference between the third term (k+1) and the second term (3k). Mathematically, we can express this as:

(3k - k) = (k + 1 - 3k)

Simplifying the equation, we get:

2k = -2k + 1

Combining like terms, we have:

4k = 1

Dividing both sides of the equation by 4, we find:

k = 1/4

Now, to find the first term (a) if 3k is the sixth term of the progression, we can use the formula for the nth term of an AP:

a + (n - 1)d = nth term

Given that the sixth term (3k) is equal to 3 times the common difference (3d), we can substitute these values into the formula:

a + (6 - 1)d = 3k

a + 5d = 3k

Since we already know that k = 1/4, we can substitute this value into the equation:

a + 5d = 3(1/4)

a + 5d = 3/4

Now, we need to find the value of d. We can use the fact that k, 3k, and k+1 are three consecutive terms of the AP:

(3k - k) = (k + 1 - 3k)

Simplifying, we have:

2k = 1 - 2k

4k = 1

k = 1/4

Since k = 1/4, we can substitute this into the equation for d:

2(1/4) = 1 - 2(1/4)

1/2 = 1 - 1/2

1/2 = 1/2

So, d = 1/2.

Substituting the values of k and d into the equation a + 5d = 3/4, we can solve for a:

a + 5(1/2) = 3/4

a + 5/2 = 3/4

Multiplying through by 4 to eliminate fractions, we have:

4a + 10 = 3

4a = 3 - 10

4a = -7

a = -7/4

Therefore, the numerical values are k = 1/4 and a = -7/4.