How is <msqrt> 2 + <msqrt> 2 + <msqrt> 2 +

Carina Valenzuela

Carina Valenzuela

Answered question

2022-04-12

How is 2 + 2 + 2 + . . . n times = 2 cos ( π / 2 n + 1 )?

Answer & Explanation

Astok3mpd

Astok3mpd

Beginner2022-04-13Added 18 answers

Recall the identity
2 cos ( θ ) = 2 + 2 cos ( 2 θ )
This means
2 cos ( θ ) = 2 + 2 + 2 cos ( 4 θ ) = 2 + 2 + 2 + 2 cos ( 8 θ )  and so on 
Setting θ = π 2 n + 1 , and going on for n terms, we see that we end up with cos ( π / 2 ), which is indeed zero.
EDIT All the square roots come with a positive sign, since each of the cosine term is of the form cos ( π / 2 k ), where k 1, i.e., π / 2 k [ 0 , π / ], where the cosine is non-negative.
Justine Webster

Justine Webster

Beginner2022-04-14Added 1 answers

2 cos ( π 2 n + 1 ) = 2 1 + cos ( π 2 n ) 2 = 2 + 2 cos ( π 2 n )
Keep on going...until you reach n=2.

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