Prove that x2 + 1/x2 = x3 + 1/x3 = x4 + 1/x4.

If x + 1/x = 2, prove that x² + 1/x² = x³ + 1/x³ = x⁴ + 1/x⁴.

Answer :

We know that

x² + 1/x² = (x + 1/x)² – 2

Substituting the values

x² + 1/x² = 2² – 2

So we get

x² + 1/x² = 4 – 2 = 2 …. (1)

x³ + 1/x³ = (x + 1/x)³ – 3 (x + 1/x)

Substituting the values

x³ + 1/x³ = 2³ – 3 × 2

So we get

x³ + 1/x³ = 8 – 6 = 2 …… (2)

x⁴ + 1/x⁴ = (x² + 1/x²)² – 2

Substituting the values

x⁴ + 1/x⁴ = 2² – 2

So we get

x⁴ + 1/x⁴ = 4 – 2 = 2 …. (3)

From equation (1), (2) and (3)

x² + 1/x² = x³ + 1/x³ = x⁴ + 1/x⁴

Hence, it is proved.

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