Let x, y, z be three positive real numbers in a geometric progression such that x < y < z. If 5x, 16y, and 12z are in an arithmetic progression then the common ratio of the geometric progression is

Let a₁, a₂, ... , a₅₂ be positive integers such that a₁ ? a₂ ? ... ? a₅₂. Suppose, their arithmetic mean is one less than the arithmetic mean of a₂, a₃, ..., a₅₂. If a₅₂ = 100, then the largest possible value of a₁ is

The value of the sum 7 x 11 + 11 x 15 + 15 x 19 + ..... + 95 x 99 is

If the population of a town is p in the beginning of any year then it becomes 3+2p in the beginning of the next year. If the population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be

Let a₁ , a₂ be integers such that a₁ – a₂ + a₃ – a₄ + ........ +(-1)^n-1 a_n = n , for n ? 1. Then a₅₁ + a₅₂ + ........ + a₁₀₂₃ equals