**Question:**

**A carpenter was hired to build 192 window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?**

**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.

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