Question

For each natural number n ³ 2, the largest possible value of the expression

S_n = sinx₁ cos x₂ + sin x₂ cos x₃ + ...+ sin x_n cos x₁ (where x₁, x₂ ... x_n are arbitrary real numbers)

• A.
• B.
• C.
• D. n

Answer 1

Answer: (B)

Answer 2

By the inequalities 2ab £ a² + b², we get

S_n £ $\frac { \sin ^ { 2 } x _ { 1 } + \cos ^ { 2 } x _ { 2 } } { 2 } + \ldots . + \frac { \sin ^ { 2 } x _ { n } + \cos ^ { 2 } x _ { 1 } } { 2 } = \frac { n } { 2 }$