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.
308 exam-style questions on OCR (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.