Trying to fit in 2016
Algebra
Level
3
The number is:
None of these choices
Real
Has modulus 1
Purely imaginary
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u = ( 1 4 + 3 i ) 2 0 1 6 + ( 3 + 1 4 i ) 2 0 1 6 = ( 1 4 + 3 i ) 2 0 1 6 + i 2 0 1 6 i 2 0 1 6 × ( 3 + 1 4 i ) 2 0 1 6 = ( 1 4 + 3 i ) 2 0 1 6 + 1 ( 3 i − 1 4 ) 2 0 1 6 = ( 1 4 + 3 i ) 2 0 1 6 + ( − 1 ) 2 0 1 6 ( 1 4 − 3 i ) 2 0 1 6 = ( 1 4 + 3 i ) 2 0 1 6 + ( 1 4 − 3 i ) 2 0 1 6 = k = 0 ∑ 2 0 1 6 ( 2 0 1 6 k ) 1 4 k ( 3 i ) 2 0 1 6 − k + k = 0 ∑ 2 0 1 6 ( − 1 ) k ( 2 0 1 6 k ) 1 4 k ( 3 i ) 2 0 1 6 − k
We note that all the k = odd terms, which are imaginary, cancel out. Therefore, u is real .