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.
the difference between consecutive terms is ... -k/3
(a) 0 - k/3 - k/3 = -2k/3
(b) k - [(n-1)k/3]
(c) from (b) ... 15 = k - [(20 - 1) * k/3] ... solve for k
To find the solutions to these questions, we need to understand the pattern in the given arithmetic progression (AP) sequence. An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant.
In this case, the given AP is k, 2k/3, k/3, 0,....
(a) Finding the sixth term:
To find the sixth term, we need to determine the common difference between consecutive terms. We can observe that the common difference is -k/3 because each term is obtained by subtracting k/3 from the previous term.
So, the sixth term can be found by adding (-k/3) five times to the first term (k):
Sixth term = k + (-k/3) * 5 = k - 5k/3 = (3k/3) - (5k/3) = -2k/3
Therefore, the sixth term is -2k/3.
(b) Finding the nth term:
To find the nth term, we need to identify the pattern in the sequence. From observation, we can see that each term can be obtained by multiplying the term number (n) by -k/3.
Therefore, the nth term of this AP can be given as: Tn = -k/3 * n
(c) Finding k given the 20th term is equal to 15:
Using the formula derived in part (b), we can substitute n = 20 and Tn = 15 into the equation to find k:
15 = -k/3 * 20
Multiplying both sides by -3:
-45 = -20k
Dividing both sides by -20:
k = -45 / -20 = 9/4
Therefore, k is equal to 9/4.