Discussion

Explanation:

Given, f(x − 1) + f(x + 1) = f(x)

We can rewrite the equation as,

f(x + 1) = f(x) – f(x – 1)

Substituting x = 1, we get

f(2) = f(1) - f(0)

⇒ 7 = f(1) – 2

⇒ f(1) = 9

∴ f(0) = 2, f(1) = 9 and f(2) = 7

Now, substituting x = 2, we get

f(3) = f(2) - f(1) = 7 – 9 = -2

Similarly,

f(4) = f(3) - f(2) = -2 – 7 = -9

f(5) = f(4) – f(3) = -9 – (-2) = -7

f(6) = f(5) – f(4) = -7 – (-9) = 2 = f(0)

f(7) = f(6) – f(5) = 2 – (-7) = 9 = f(1)

f(8) = (7) – f(6) = 9 – 2 = 7 = f(2)

Hence, value of f(x) repeats after every 6 integral value of x.

∴ f(6x) = f(0) = 2
f(6x + 1) = f(1) = 9,
f(6x + 2) = f(2) = 7,
f(6x + 3) = f(3) = -2,
f(6x + 4) = f(4) = -9,
f(6x + 5) = f(5) = -7

Now, f(1000) = f(6 × 166 + 4) = f(4) = -9

Hence, option (c).

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