If g(x) = g(x - 1) × g(x + 1) and g(0) = 1/3 and g(5)= 1/6, then g(1 + 2 + 3 + … + 100) =
Explanation:
Given, g(x) = g(x - 1) × g(x + 1) and g(0) = 1/3.
⇒ g(x + 1) = g(x)/g(x - 1)
Suppose g(1) = a
Put x = 1, we get g(2) = a/(1/3) = 3a.
Put x = 2, we get g(3) = 3a/a = 3.
Put x = 3, we get g(4) = 3/3a = 1/a
Put x = 4, we get g(5) = (1/a)/3 = 1/3a
Also it is given, g(5) = 1/6
∴ 1/3a = 1/6
⇒ a = 2.
∴ g(4) = 1/2 and g(5) = 1/6
Put x = 5, we get g(6) = (1/6)/(1/2) = 1/3 = g(0)
Put x = 6, we get g(7) = (1/3)/(1/6) = 2 = g(1)
Hence, we can see that g(6 + x) = g(x).
∴ g(1 + 2 + 3 + … + 100) = g(5050) = g(6 × 841 + 4) = g(4) = ½
Hence, option (b).
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