Assume 2 + 2² + 2³ + . . . + 2^k = k² , then prove the sum formula.

Assume the statement 2 + 2² + 2³ + . . . + 2^k = 2^{k + 1} – 2, then show it holds for k + 1.

Assume the statement is true for k, then show it is true for n.

Assume the statement 2 + 2² + 2³ + . . . + 2^k = 2^{k + 1} , then show it holds for k – 1.

2+2i      

 1 – i

 1 – 3i

   2–2i      

x sin x  + 2 cos x

− x sin x  –  2 cos x

− x sinx  + 2 cos x

x sin x  –  2 cos x

9

8

13

5

9

3

0

6

81

71

48

85

1  

4  

2  

3  

9

4

3

5