The total energy E(n)E(n)E(n) released by a sequence of n n\,n laser pulses in a laboratory experiment is given by the sum E(n)=∑j=0n(2j)4E(n) = \sum_{j=0}^{n} (2j)^4E(n)=∑j=0n(2j)4 millijoules, where n n\,n is a positive integer. The specific energy produced by the nnn-th pulse is defined as P(n)=E(n)−E(n−1)P(n) = E(n) - E(n-1)P(n)=E(n)−E(n−1).
Determine the specific energy released by the 3rd pulse, P(3)P(3)P(3), and the 10th pulse, P(10)P(10)P(10).
Find the pulse number n n\,n such that the specific energy released is P(n)=8.1×109P(n) = 8.1 \times 10^9P(n)=8.1×109 millijoules.