An Ap is given by K, 2k/3,K/3,0.... A) find the sixth term? B) find the nth term? C) if the 20th term is equal to 15, find K.?

A) To find the sixth term, we can use the formula for the nth term of an AP:
Tn = a + (n-1)d
where Tn is the nth term, a is the first term, n is the term number, and d is the common difference.

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

So, the sixth term would be:
T6 = K + (6-1)(-K/3)
T6 = K - 5K/3
T6 = -2K/3

Therefore, the sixth term is -2K/3.

B) To find the nth term, we can use the same formula:
Tn = K + (n-1)(-K/3)
Tn = K - (n-1)K/3
Tn = K - nK/3 + K/3
Tn = K(1 - n/3) + K/3
Tn = K(3-n)/3

Therefore, the nth term is K(3-n)/3.

C) Given that the 20th term is equal to 15, we can substitute n = 20 into the nth term formula and set it equal to 15:
15 = K(3-20)/3
15 = K(-17)/3
15 = -17K/3
45 = -17K
K = -45/17

Therefore, K = -45/17.

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