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`        If f(x-1)+f(x+1)=√3f(x) and f(5)=100 for all x€R then find the period of f(x)`
3 years ago Nishant Vora
IIT Patna
2467 Points
```							Replace x = (x + 2) in above equation,=> ƒ(x + 2 + 2) + ƒ(x + 2 − 2) = (√2) * ƒ(x + 2)=> ƒ(x + 4) + ƒ(x) = (√2) * ƒ(x + 2) - - - (1)Again replace x = (x − 2) in above equation,=> ƒ(x − 2 + 2) + ƒ(x − 2 − 2) = (√2) * ƒ(x − 2)=> ƒ(x) + ƒ(x − 4) = (√2) * ƒ(x − 2) - - - (2)Adding (1) & (2) we get,=> ƒ(x + 4) + ƒ(x − 4) + 2ƒ(x) = (√2) * ƒ(x + 2) + (√2) * ƒ(x − 2)=> ƒ(x + 4) + ƒ(x − 4) + 2ƒ(x) = (√2) * [ƒ(x + 2) + ƒ(x − 2)]=> ƒ(x + 4) + ƒ(x − 4) + 2ƒ(x) = (√2) * (√2) ƒ(x ). . . . . . . .{As, ƒ(x + 2) + ƒ(x − 2) = (√2) * ƒ(x) }=> ƒ(x + 4) + ƒ(x − 4) + 2ƒ(x) = 2 ƒ(x )=> ƒ(x + 4) + ƒ(x − 4) = 0Now replace, x = (x + 4),=> ƒ(x + 8) + ƒ(x) = 0=> ƒ(x + 8) = −ƒ(x) - - - - - - (3)Replacing x = (x + 8),=> ƒ(x + 16) = −ƒ(x + 8)=> ƒ(x + 16) = − [−ƒ(x)]=> ƒ(x + 16) = ƒ(x)
```
3 years ago
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