In a geometric progression, the sum of the first and the last terms is 66 and the product of the second and the last second terms is 128. Determine the first term of the series?

2508 ÷ 15.02 ÷ ? x 11 = 200
? 2508/15 + ? x 11 = 200
? ? = (200 - 167.2) x 1/11 = 2.98 ? 3

Upstream speed of boat = Boat's speed in still water - Rate of stream = 4 - 2 = 2 km/h

Let the requirement odd integers be N, N +2 and N + 4
Then, N + (N + 2) + (N + 4) = 57
? 3N = 51
? N = 17
? The integers are 17, 19, 21.

Because of stoppages, train covers 50 km less per hour.
? Time taken taken to cover 50 km = 50/150 h = (50/150) x 60 = 20 min

Let age of Raman's daughter = N yr
Then, age of Raman's = 3N yr
and age of Raman's mother = (13/9) x 3N = 13N/3 yr
According to the question, N + 3N + 13N/3 = 125
? 3N + 9N + 13N = 125 x 3
? N = (125 x 3)/25 = 15
Hence, required difference = 13N/3 - N = 10N/3
= (10 x 15)/3 = 50 yr

Let boat's rate upstream be U and boat's rate downstream = V

According to the question.
Distance covered in 528 min = Distance covered in 240 min
? Distance covered in 8 h 48 min = Distance covered in 4 h
? U x 8⁴/₅ = V x 4
? (44 x U)/5 = 4V
? V = (11 x U)/5
? Required ratio = (V + U)/2 : (V - U)/2
? = (11U/5 + U)/2 : (11U/5 - U)/ 2
? = 16U/5 : 6U/5 = 8 : 3

Let A's age 10 years ago = x years
Then, B's age 10 years ago = 2x years
? (x + 10) / (2x + 10) = 3 / 4
? 4(x + 10) = 3(2x + 10)
? x(6 - 4) = 40 - 30
? x = 5

&ther4; Total of their present ages = (x + 10) + (2x + 10)
= (3x + 20)
= 35 years

Let B's present age be x years then A's present age be (x + 9) years
As per the given condition
(x + 9 + 10) = 2(x - 10)
? x = 39

? The present age of B = 39 years

75/100 = 0.75