A carpenter was hired to build 192 window frames.

Solution:

Given first day he made 5 frames then two frames more than the previous that is 7 then 9 and so on

Hence the sequence of making frames each day is 5, 7, 9…

The sequence is AP with first term as a = 5 and common difference d = 2

Total number of frames to be made is 192

Let is requires n days hence Sn = 192

The sum of first n terms of AP is given by Sn = n/2 (2a + (n – 1) d)

Where a is the first term and d is common difference

Sn = (n/2) (2(5) + (n – 1) 2)

192 = (n/2) (10 + 2n – 2)

⇒ 384 = 10n + 2n2 – 2n

On computing and simplifying we get

⇒ 2n2 + 8n – 384 = 0

⇒ n2 + 4n – 192 = 0

⇒ n2 + 16n – 12n – 192 = 0

⇒ n (n + 16) – 12 (n + 16) = 0

⇒ (n – 12) (n + 16) = 0

⇒ n = 12 and n = -16

But n represents number of days which cannot be negative hence n = 12

Hence number of days required to finish the job is 12 days.