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Indukcja matematyczna: 

1.

 

2

)

1

(

...

3

2

1

+

=

+

+

+

+

n

n

n

 

2.

 

(

)

(

)

2

2

3

3

3

3

2

1

...

3

2

1

...

3

2

1





+

=

+

+

+

+

=

+

+

+

+

n

n

n

n

 

3.

 

1

)

1

(

1

...

4

3

1

3

2

1

2

1

1

+

=

+

+

+

+

+

n

n

n

n

 

4.

 

)

1

(

2

2

)

1

(

1

1

...

9

1

1

4

1

1

2

+

+

=





+

n

n

n

 

5.

 

2

1

3

3

...

3

3

1

2

1

=

+

+

+

+

n

n

 

6.

 

n

n

n

n

2

2

2

2

...

2

3

2

2

2

1

3

2

+

=

+

+

+

+

 

7.

 

(

)

(

)(

)(

)

5

3

2

1

12

1

1

...

4

2

3

2

2

1

2

2

2

2

+

+

+

=

+

+

+

+

+

n

n

n

n

n

n

 

8.

 

(

) (

)

1

2

3

4

...

9

5

1

=

+

+

+

+

n

n

n

 

9.

 

(

) (

)

1

3

2

6

...

16

10

4

+

=

+

+

+

+

n

n

n

 

10.

 

( )

6

2

3

1

...

4

3

2

1

2

3

2

1

2

2

2

2

n

n

n

n

n

+

+

=

+

+

+

 

11.

 

(

)

(

)

(

)

3

8

4

1

3

1

1

2

...

5

3

1

2

2

2

2

2

+

+

+

=

+

+

+

+

+

n

n

n

n

 

12.

 

0

)

1

(

,

1

1

...

1

1

1

1

1

4

3

2

1

}

1

{

\

+

=

=

+

+

+

+

n

n

n

n

n

a

n

r

a

a

a

a

n

a

a

a

a

a

a

N

n

 

13.

 

(

)

10

9

10

81

1

1

...

11

...

111

11

1

1

=

+

+

+

+

+

n

n

jedynek

n

 

14.

 

2

sin

2

2

1

cos

2

cos

sin

...

3

sin

2

sin

sin

x

x

n

x

nx

x

x

x





+

=

+

+

+

+

 

15.

 

(

)

1

1

...

}

1

{

\

3

2

=

+

+

+

+

+

a

a

a

a

a

a

a

R

a

N

n

n

n

 

16.

 

(

)

(

)

)

2

(

1

3

1

1

...

4

3

3

2

2

1

+

+

=

+

+

+

+

+

n

n

n

n

n

 

17.

 

(

)(

)

1

3

1

3

2

3

1

...

7

4

1

4

1

1

+

=

+

+

+

+

n

n

n

n

 

18.

 

( )

(

)

(

)

2

2

2

3

3

3

1

2

2

...

4

2

2

2

...

4

2

+

=

+

+

=

+

+

+

n

n

n

n

 

19.

 

1

)!

1

(

!

...

!

3

3

!

2

2

!

1

1

+

=

+

+

+

+

n

n

n