150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.
Let total work = 1
and let total work be completed in 'n' days
Work done in 1 day = 1/n
This is the work done by 150 workers
Work done by 1 worker in one day = 1/150n
Case 1: -
No. of workers = 150
Work done per worker in 1 day = 1/150n
Total work done in 1 day = 150/150n
Case 2: -
No. of workers = 146
Work done per worker in 1 day = 1/150n
Total work done in 1 day = 146/150n
Case 3: -
No. of workers = 142
Work done per worker in 1 day = 1/150n
Total work done in 1 day = 142/150n
Given that
In this manner it took 8 more days to finish the work i.e. work finished in (n + 8) days.
∴ 
…(1)
Now,
is an AP
where,
first term(a) = 150
common difference(d) = 146 - 150 = - 4
we know that
Sum of n terms of AP(Sn) = (n/2)[2a + (n - 1)d]
putting n = n + 8, a = 150 & d = - 4
Sn + 8 = [(n + 8)/2] × [2(150) + (n + 8 - 1)( - 4)]
= [(n + 8)/2] × [300 + (n + 7)( - 4)]
= [(n + 8)/2] × [300 - 4n - 28]
= [(n + 8)/2] × [272 - 4n]
= (n + 8) × (136 - 2n)
= - 2n2 + 120n + 1088
From (1),
Sn + 8 = 150n
⇒ - 2n2 + 120n + 1088 = 150n
⇒ - 2n2 - 30n + 1088 = 0
⇒ - 2(n2 + 15n - 544) = 0
⇒ (n2 + 15n - 544) = 0
⇒ n2 + 32n - 17n - 544 = 0
⇒ n(n + 32) - 17(n + 32) = 0
⇒ (n - 17)(n + 32) = 0
∴ n = 17
because n = - 32 is invalid as no. of days can't be - ve.
Hence, n =17
Thus, the work was completed in n + 8 days i.e. 17 + 8 = 25 days