Difference between revisions of "1977 AHSME Problems/Problem 16"
(Created page with "== Problem 16 == If <math>i^2 = -1</math>, then the sum <cmath>\cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} + \cdots + i^{40}\cos{3645^\circ} </cma...") |
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Solution by e_power_pi_times_i | Solution by e_power_pi_times_i | ||
− | Notice that the sequence repeats every <math>4</math> terms. These <math>4</math> terms are <math>\cos(45^\circ)</math>, <math>i\cos(135^\circ)</math>, <math>-\cos(225^\circ)</math>, and <math>-i\cos(315^\circ)</math>. This is because <math>\cos(\theta) = \cos(360^\circ - \theta)</math>. Also <math>\cos(\theta) = -\cos(180^\circ - \theta)</math>, so the <math>4</math> terms are just <math>\cos(45^\circ)</math>, <math>-i\cos(45^\circ)</math>, <math>\cos(45^\circ)</math>, and <math>-i\cos(45^\circ)</math>, which adds up to <math>\sqrt{2}-i\sqrt{2}</math>. The sequence repeats <math>10</math> times, and an extra <math>i^ | + | Notice that the sequence repeats every <math>4</math> terms. These <math>4</math> terms are <math>\cos(45^\circ)</math>, <math>i\cos(135^\circ)</math>, <math>-\cos(225^\circ)</math>, and <math>-i\cos(315^\circ)</math>. This is because <math>\cos(\theta) = \cos(360^\circ - \theta)</math>. Also <math>\cos(\theta) = -\cos(180^\circ - \theta)</math>, so the <math>4</math> terms are just <math>\cos(45^\circ)</math>, <math>-i\cos(45^\circ)</math>, <math>\cos(45^\circ)</math>, and <math>-i\cos(45^\circ)</math>, which adds up to <math>\sqrt{2}-i\sqrt{2}</math>. The sequence repeats <math>10</math> times, and an extra <math>i^{40}\cos(3645^\circ)</math>. The sum is <math>10(\sqrt{2}-i\sqrt{2})+\dfrac{\sqrt{2}}{2} = \dfrac{21\sqrt{2}}{2} - 10i\sqrt{2} = \boxed{\text{(D) }\dfrac{\sqrt{2}}{2}(21-20i)}</math>. |
Latest revision as of 11:52, 22 November 2016
Problem 16
If , then the sum equals
Solution
Solution by e_power_pi_times_i
Notice that the sequence repeats every terms. These terms are , , , and . This is because . Also , so the terms are just , , , and , which adds up to . The sequence repeats times, and an extra . The sum is .