Penalty for the first day (a) = Rs 4000
Increased rate for every day (d) = Rs 1000
Maximum amount of penalty (Sn)
= Rs 1,65,000
Sn = (n/2) [2a + (n - 1) 1000]
165000 = (n/2) [2(4000) + (n – 1) 1000]
= (n/2) [8000 + 1000 n – 1000]
165000 = (n/2) (7000 + 1000n)
330000 = 7000 n + 1000 n2
0 = 1000 n2 + 7000 n – 330000 ÷ 1000 on both sides,
n2 + 7n – 330 = 0
(n + 22) (n – 15) = 0
n = -22 or n = 15
n = -22 (not possible)
∴ Maximum number of days for which the work can be delayed is 15 days.