Difference between revisions of "1957 AHSME Problems/Problem 42"
Brendanb4321 (talk | contribs) (Created page with "== Problem 42== If <math>S = i^n + i^{-n}</math>, where <math>i = \sqrt{-1}</math> and <math>n</math> is an integer, then the total number of possible distinct values for <m...") |
m (→Solution) |
||
(One intermediate revision by one other user not shown) | |||
Line 7: | Line 7: | ||
==Solution== | ==Solution== | ||
− | We first use the fact that <math>i^{-n}=\frac1{i^n}=\left(\frac1i\right)^n=(-i)^n</math>. Note that <math>i^4=1</math> and <math>(-i)^4=1</math>, so <math>i^n</math> and <math>(-i)^n</math> | + | We first use the fact that <math>i^{-n}=\frac1{i^n}=\left(\frac1i\right)^n=(-i)^n</math>. Note that <math>i^4=1</math> and <math>(-i)^4=1</math>, so <math>i^n</math> and <math>(-i)^n</math> are periodic with periods at most 4. Therefore, it suffices to check for <math>n=0,1,2,3</math>. |
Line 19: | Line 19: | ||
Hence, the answer is <math>\boxed{\textbf{(C)}\ 3}</math>. | Hence, the answer is <math>\boxed{\textbf{(C)}\ 3}</math>. | ||
+ | |||
+ | == Solution 2 == | ||
+ | |||
+ | Notice that the powers of <math>i</math> cycle in cycles of 4. So let's see if <math>S</math> is periodic. | ||
+ | |||
+ | For <math>n=0</math>: we have <math>2</math>. | ||
+ | |||
+ | For <math>n=1</math>: we have <math>0</math>. | ||
+ | |||
+ | For <math>n=2</math>: we have <math>-2</math>. | ||
+ | |||
+ | For <math>n=3</math>: we have <math>0</math>. | ||
+ | |||
+ | For <math>n=4</math>: we have <math>2</math> again. Well, it can be seen that <math>S</math> cycles in periods of 4. Select <math>\boxed{C}</math>. | ||
+ | |||
+ | ~hastapasta |
Revision as of 14:47, 10 June 2024
Problem 42
If , where and is an integer, then the total number of possible distinct values for is:
Solution
We first use the fact that . Note that and , so and are periodic with periods at most 4. Therefore, it suffices to check for .
For , we have .
For , we have .
For , we have .
For , we have .
Hence, the answer is .
Solution 2
Notice that the powers of cycle in cycles of 4. So let's see if is periodic.
For : we have .
For : we have .
For : we have .
For : we have .
For : we have again. Well, it can be seen that cycles in periods of 4. Select .
~hastapasta