Answer:
f(1986) = 0
Step-by-step explanation:
f(a + b) = f(a) + f(b) - 2f(ab)
We need to find f(1986)
f(1986) = f(1 +1985). Using the above formula, we can write:
f(1 +1985) = f(1) + f(1985) - 2f(1 x 1985)
f(1986) = 1 - f(1985) Equation 1
Applying the same formula again on f(1985), we get:
f(1985) = f(1 + 1984) = f(1) + f(1984) - 2f(1984)
f(1985) = 1 - f(1984)
Using this value in Equation 1, we get:
f(1986) = 1 - (1 - f(1984))
f(1986)= f(1984)
Continuing this, we can observe,
f(1986) = f(1984) = f(1982) = f(1980) .... = f(4) = f(2)
So,
f(1986) = f(2)
f(2) = f(1 + 1) = f(1) +f(1) - 2(1 x 1) = f(1) + f(1) - 2f(1)
f(2) = 1 + 1 - 2 = 0
Therefore,
f(1986) = f(2) = 0
