1964 AHSME Problems/Problem 34
Problem
If is a multiple of , the sum , where , equals:
Solution
The real part is , which is . If is a multiple of , then we have an odd number of terms in total: we start with at , then add two more terms to get at , etc. With each successive addition, we're really adding a total of , since , and , etc.
At , the sum is , and at , the sum is . Since the sum increases linearly, the real part of the sum is .
The imaginary part is , which is . This time, we have an even number of terms. We group pairs of terms to get , and notice that each pair gives . Again, with the imaginary part is , while with the imaginary part is . Since again the sum increases linearly, this means the imaginary part is .
Combining the real and imaginary parts gives , which is equivalent to option .
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 33 |
Followed by Problem 35 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.