Question

1. Show that, for every n > 1: n ka n(n + 1)(2n +1) 6 k=1

Answer 1

n n(n+1)(2041) K=1 FOR L.H.S I= 1(1+1)(2+0 RoHS 6 1. for hold. FOR n=2 L.H.S 5 R. HIS 30 212+1)(4+1) 6 = 5 for n = 2 0 ý hold, Let y assume for n=p, 6 ý hold, E K P(+1) (22+1) K=1 6 Y 41 NOW È x + (8 + 152 K=1 Kal from r(0+1)(22+1) 6 ++)"

-->{23 (n+1){ 20+70+6} 11 6840) { 20(*+2)+3(v+2) principal of induction theorem Now (M+1) { 20+ pt 6P+6 6 (+) { 287 49 + 30+6 G 60+1 6 (M+ 1) (2+1+1). +(2(8+1)+ I 6. for nart1, o ý hold herefore by The relation Σ κ' x² nc n+1) (2n+1) x=1 6 11 hold for all natural number (ny, 1).