The volume of telemetry data, DDD, in petabytes, captured by a deep-space relay station is observed over its first three years of operation. The data volumes for years 1, 2, and 3 are modeled as the first three terms of a geometric series and are given by
2k−15,k,k+10 2k - 15, \quad k, \quad k + 10 2k−15,k,k+10where kkk is a constant.
Prove that k2+5k−150=0k^2 + 5k - 150 = 0k2+5k−150=0.
Determine the volume of data predicted to be captured in year 4.
The cumulative volume of data captured since the start of operation is predicted to exceed 5×1065 \times 10^65×106 petabytes for the first time in year NNN. Calculate the value of NNN.
