Question 3 - A-Level Maths

Exam Board	OCR MEI
Module	Further Pure Core AS (Further Pure Core AS)
Year	2024
Session	June
Topic	Sequences and Series

Using standard summation formulae, write down an expression in terms of \(n\) for \(\sum _ { r = 1 } ^ { 2 n } r ^ { 3 }\).
Hence show that \(\sum _ { \mathrm { r } = \mathrm { n } + 1 } ^ { 2 \mathrm { n } } \mathrm { r } ^ { 3 } = \frac { 1 } { 4 } \mathrm { n } ^ { 2 } ( \mathrm { an } + \mathrm { b } ) ( \mathrm { cn } + \mathrm { d } )\), where \(a , b , c\) and \(d\) are integers to be determined.