A modular structural analysis of a skyscraper uses a load-bearing index L(m)L(m)L(m) for the mmm-th floor. This index is calculated by the difference between two cumulative load sums:
L(m)=∑k=0m5k4−∑k=0m−15k4 L(m) = \sum_{k=0}^{m} 5k^4 - \sum_{k=0}^{m-1} 5k^4 L(m)=k=0∑m5k4−k=0∑m−15k4where mmm is a positive integer representing the floor number.
Find the load-bearing index for the 4th floor, L(4)L(4)L(4), and the 10th floor, L(10)L(10)L(10).
An engineer determines that the maximum theoretical limit for a floor's load-bearing index is 3.125×10113.125 \times 10^{11}3.125×1011. Calculate the floor number mmm that reaches this limit.
308 exam-style questions on OCR A Level Maths 1.4 Sequences and Series, covering 1.4.1 Binomial expansion for positive integer n, 1.4.2 Link to binomial probabilities, 1.4.3 Binomial expansion for rational n (A-level only), 1.4.4 Validity of the expansion (A-level only), 1.4.5 Sequences (A-level only), 1.4.6 Increasing, decreasing and periodic sequences (A-level only), 1.4.7 Sigma notation (A-level only), 1.4.8 Arithmetic sequences and series (A-level only), 1.4.9 Geometric sequences and series (A-level only), 1.4.10 Sum to infinity of a geometric series (A-level only), and 1.4.11 Modelling with sequences and series (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.