AQA
Further AS Paper 1
2023
June
— Question 13
4 marks
Exam Board
AQA
Module
Further AS Paper 1 (Further AS Paper 1)
Year
2023
Session
June
Marks
4
Topic
Proof by induction
13
Prove by induction that, for all integers \(n \geq 1\),
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$
[4 marks]
13
Hence, or otherwise, write down a factorised expression for the sum of the first \(2 n\) squares
$$1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$
13
Use the formula in part (a) to write down a factorised expression for the sum of the first \(n\) even squares
$$2 ^ { 2 } + 4 ^ { 2 } + 6 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$
13
Hence, or otherwise, show that the sum of the first \(n\) odd squares is
$$a n ( b n - 1 ) ( b n + 1 )$$
where \(a\) and \(b\) are rational numbers to be determined.