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Question
Which term of the A.P. `20, 19 1/4, 18 1/2, 17 3/4,` ... is the first negative term?
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Solution 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 nth term of the given A.P. be the first negative term. Then,
an < 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 28th term is the first negative term of the given A.P.
Solution 2
Here, a = 20
And d = `77/4 - 20 = - 3/4`
Let tn < 0
∵ tn = 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.
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