An AP is given by K 2k/3, k/3, 0, .... a) Find the sixth term b) Find the n th term c) If the 20th term is equal to 15, find k.?

a) To find the sixth term, we need to substitute n = 6 into the formula for the nth term of an AP:

Tn = K + (n-1)d

where Tn is the nth term, K is the first term, n is the term number, and d is the common difference.

In this case, the first term K = 2k/3 and the common difference d = k/3.

So, the sixth term is:

T6 = 2k/3 + (6-1)(k/3)
T6 = 2k/3 + 5k/3
T6 = 7k/3

Therefore, the sixth term is 7k/3.

b) The nth term of an AP is given by:

Tn = K + (n-1)d

So, the nth term of this AP is:

Tn = 2k/3 + (n-1)(k/3)
Tn = 2k/3 + nk/3 - k/3
Tn = (2 + n - 1)k/3
Tn = (n + 1)k/3

Therefore, the nth term of this AP is (n + 1)k/3.

c) Given that the 20th term is equal to 15, we can set up the equation:

T20 = 15
(20 + 1)k/3 = 15
21k/3 = 15
7k = 45
k = 45/7

Therefore, k = 45/7.

a) Sixth term

The common difference, d = (2k/3 - k/3) = k/3.

To find the sixth term, we use the formula:

Sixth term = First term + (n - 1)d

Here, n = 6 (since we want the sixth term) and the first term is K.

Therefore, Sixth term = K + (6 - 1) - (k/3) = K + 5k/3.

Sixth term = 5/3 - K

b) n th term

The n th term of an AP is given by:

n th term = First term + (n - 1)d

Here, the first term is K and the common difference is k/3.

Therefore, n th term = K + (n - 1) - (k/3).

n th term = K + (n - 1)k/3

c) If the 20th term is equal to 15

Given that the 20th term is 15, we have:

20th term = K + (20 - 1) - (k/3) = 15

Simplifying:

K + 19k/3 = 15

Multiplying both sides by 3:

3K + 19k = 45

Combining like terms:

22k = 45

Therefore, k = 45/22 = 15/7.