If F k = 1 + s i n π 2 k 1 + sin ⁡ k - 1 π 2 k 1 + sin ⁡ 2 k + 1 π 2 k 1 + sin ⁡ 3 k - 1 π 2 k , then the value of F 1 + F 2 + F 3 is equal to

Question

If F k = 1 + s i n π 2 k 1 + sin ⁡ k - 1 π 2 k 1 + sin ⁡ 2 k + 1 π 2 k 1 + sin ⁡ 3 k - 1 π 2 k , then the value of F 1 + F 2 + F 3 is equal to, 3 16, 1 4, 5 16, 7 16

MARKS App · Accepted Answer

F 1 = 1 + s i n π 2 1 + s i n 0 1 + s i n 3 π 2 1 + sin ⁡ π = 2 × 1 × 0 × 1 = 0 F 2 = 1 + s i n π 4 1 + s i n π 4 1 + s i n 5 π 4 1 + s i n 5 π 4 = 1 + 1 2 2 1 - 1 2 2 = 1 - 1 2 2 = 1 4 F 3 = 1 + s i n π 6 1 + s i n π 3 1 + s i n 7 π 6 1 + s i n 4 π 3 = 1 + 1 2 1 + 3 2 1 - 1 2 1 - 3 2 = 1 - 1 4 1 - 3 4 = 3 4 × 1 4 = 3 16 ⇒ F 1 + F 2 + F 3 = 0 + 1 4 + 3 16 = 7 16

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