Question

Use mathematical induction to prove that the following statement is true for every positive integer n

Answer 1

to prove

8

+

16

+

24

+

...

+

8

n

=

4

n

(

n

+

1

)

−

−

−

−

(

*

)

let  

T

n

=

4

n

(

n

+

1

)

(1) verify for  

n

=

1

L

H

S

=

8

R

H

S

=

4

×

1

(

1

+

1

)

=

4

×

2

=

8

∴

true for  

n

=

1

# to show

T

k

⇒

T

k

+

1

assume true for  

T

k

=

4

k

(

k

+

1

)

need to show

T

k

+

1

=

4

(

k

+

1

)

(

k

+

2

)

add next term to to both sides of  

(

*

)

8

+

16

+

24

+

...

+

8

k

+

8

(

k

+

1

)

=

4

k

(

k

+

1

)

+

8

(

k

+

1

)

∴

T

k

+

1

=

4

k

(

k

+

1

)

+

8

(

k

+

1

)

=

4

(

k

+

1

)

[

k

+

2

]

=

T

k

+

1

i

.

e

.

T

k

⇒

T

k

+

1

as required

#(3) conclusion

statement true for  

T

1

∵

T

k

⇒

T

k

+

1

T

1

⇒

T

2

⇒

T

3

⇒

...

∀

n

∈

N