Question

Which term of the A.P. `20, 19 1/4, 18 1/2, 17 3/4,` ... is the first negative term?

Answer 1

The given A.P. is `20, 19 1/4, 18 1/2, 17 3/4,` ...

Here, a = 20

And d = `19 1/4 - 20`

= `77/4 - 20`

= `(77 - 80)/4`

= `-3/4`

Let the n^th term of the given A.P. be the first negative term. Then,

a_n < 0 

⇒ `20 + (n - 1) xx (-3/4) < 0`   ...[ a_n = a + (n – 1) d]

⇒ `20 + 3/4 - 3/4 n < 0`

⇒ `83/4 - 3/4 n < 0`

⇒ `-3/4 n < - 83/4`

⇒ `n > 83/3 = 27 2/3`

∴ n = 28

Hence, the 28^th  term is the first negative term of the given A.P.

Answer 2

Here, a = 20

And d = `77/4 - 20 = - 3/4`

Let t_n < 0

∵ t_n = a + (n – 1)d

∴ `20 + (n - 1) (- 3/4) < 0`

⇒ 80 – 3n + 3 < 0

⇒ 83 – 3n < 0

⇒ `n > 83/3`

⇒ n > 27.6

⇒ n = 28

Hence, the first negative term is 28.