Math, asked by Esha058, 5 months ago

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

Answers

Answered by Yashicaruthvik
1

Answer:

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,...#

1

+

2

+

3

+

.

.

n

=

1

2

n

(

n

+

1

)

 

(

1

)

verify for  

n

=

1

L

H

S

=

1

R

H

S

=

1

2

×

1

×

(

1

+

1

)

=

1

2

×

1

×

2

=

1

true for  

n

=

1

(

2

)

to prove  

T

k

T

k

+

1

assume true for  

T

k

=

1

2

k

(

k

+

1

)

to prove  

T

k

+

1

=

1

2

(

k

+

1

)

(

k

+

2

)

add the next term

R

H

S

=

1

2

k

(

k

+

1

)

+

(

k

+

1

)

=

(

k

+

1

)

(

1

2

k

+

1

)

=

1

2

(

k

+

1

)

(

k

+

2

)

=

T

k

+

1

 as required

T

k

T

k

+

1

(

3

)

conclusion

(

i

)

 

T

1

is true

(

i

i

)

 

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  

n

N

Step-by-step explanation:

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