Prove by mathematical induction that 1+2+3......+n=1/2n(n+1)?

1 Answer
Apr 15, 2018

see below

Explanation:

to prove by induction

#1+2+3+..n=1/2n(n+1)#

#color(red)((1) " verify for " n=1)#

#LHS=1#

#RHS=1/2xx1xx(1+1)=1/2xx1xx2=1#

#:. "true for "n=1#

#color(red)((2)" to prove "T_k=>T_(k+1))#

#"assume true for "T_k=1/2k(k+1)#

to prove #T_(k+1)=1/2(k+1)(k+2)#

add the next term

#RHS=1/2k(k+1)+(k+1)#

#=(k+1)(1/2k+1)#

#=1/2(k+1)(k+2)=T_(k+1)" as required"#

#:. T_k=>T_(k+1)#

#color(red)((3) " conclusion"#

#(i) " "T_1" is true"#

#(ii)" " T_k=>T_(k+1)#

#:. T_1=>T_2#

#T_2=>T_3" etc."#

therefore by induction true for values

#1,2,3,...#

and so #AAninNN#