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Sum

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?

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#### Solution

Here, first term a = 5 and the common difference d = 2

Let the carpenter will take n days to finish the job

S_{n} = 192

S_{n} = `n/2[2a + (n - 1)d]`

192 = `n/2[2 xx 5 + (n - 1)2]`

⇒ 192 × 2 = n[10 + 2n – 2]

⇒ 384 = n(2n + 8)

⇒ 384 = 2n^{2} + 8n

⇒ 2n^{2} + 8n – 384 = 0

⇒ n^{2} + 4n – 192 = 0

⇒ n^{2} + 16n – 12n – 192 = 0

⇒ n(n + 16) – 12(n + 16) = 0

⇒ (n – 12)(n + 16) = 0

⇒ n = 12 ....[∵ n ≠ –16]

Hence, the required number of days = 12.

Concept: Concept of Series

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