Q) Which term of the A.P. : 65, 61, 57, 53,………… is the first negative term.

Ans: This is A.P. of decreasing order. In the given A.P., we can see that:

First term a = 65 and common difference d = -4

Let first negative term be n^th term, say T_n

We know that n^th term of an A.P. is given by T_n =  a + (n-1) d

Since it has to be first negative term, therefore:

a + (n-1) d < 0

   65 + (n-1) (-4) < 0

69 – 4n < 0

69 < 4n

n > = 17.25

Since term count has to be integer, hence, nearest integer value which is greater than 17.25 is 18.

Therefore, 18th term of the given A.P. will be its first negative term.

Check:

Value of 17^th term will be, N₁₇ = 65 + (17 – 1) (- 4) = 65 – 64 = 1

and value of 18^th term will be, N₁₈ = 65 + (18 – 1) (- 4) = 65 – 68 = – 3, which will be the first negative term after 1.