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 AQA A Level Maths 1.7 D: Sequences and series, covering 1.7.1 Binomial expansion, 1.7.2 Types of sequence (A-level only), 1.7.3 Sigma notation (A-level only), 1.7.4 Arithmetic sequences and series (A-level only), 1.7.5 Geometric sequences and series (A-level only), and 1.7.6 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.