Question

use mathematical induction to prove the following

* n(n+1)(n+2) 34 + 1) = n(n + y(n = 3). 2* = 2n+1 – 1. (4k + 1) = (n + 1)(2n + 1). k=0

Answer 1

To prove by mahemad cal in dec ton-on n. 3 Jar n LHS (Kt) l.2 2 1+ 0u2)2.3 R.H.S z 2 3 3 .LH.SER.HS LH.S (k11) 12+ 2.3 2+6-8 k리 2/2+)(2r2) R.HS 2x34 8 3 3 . LHS . H-S Then, Dis mue for n-2 Suppose, Dis true or n-m Then, ) mm)(m) 3 Noay We have to shoa thal, is true Jor n- mt mti Naa, K)2)+m)() K21 m 3 m 3 3 imLtmar)+1) ina)+1) S, is toue or

Then is true for all n by Mathematical induction To prove Σρκ. nti for m=1, L.H.S 22+2'3 R.H.S 2-1 3 L20 LHSEP.HS r n-2, LHS-É2,2227, 2 RHS.2-1-7 So, is true for n-1,2 Juppose, is h ue or n2 m mt1 Then, 2 २१ 2 2 - / Nou, We have do show that is tmuefor name) K 2 K-U 2 -+2 1+uu 2.2mt1 - / mt2 2 -2 2 - / So toue for nmel he n foue for All n by 2 2 - / 20 Mathematical induehon

t)(n1) (2n1)O Z1x4)> 14(44)> 51LHS-6 or nel, 41) (2+1) > 243. 6. R.H.S 2.W.S R.HS 2 M(4Kt1 1+5+9-15 LHS=15 20 L1)(2211)3x5 - 15 p.H.S .LHS R.HS Io, i true Jar n-1 2 Juppese, iA true for n-m (1t) -(me)/2m+) m K-O We have to show Dis true dor = mt/ mt Σ(1-) k0 (4Kt)11) # Cn-11 (2mt)+ 1m t5 7 7 2m t 3m+4m+1t5 2m2t 7m + 6 2m+4mt)mt 6 2m (mt2) t 3(mt2) Jo, is true for n-m in tue for all n by indue ton (4K)-)