# The Lucas numbers L_n are defined by the equations L_1 = 1 and L_n = F_(n+1) + F_(n-1) for each n > or equal to 2. Prove that L_1 + 2L_2 + 4L_3 +8L_4 + ... + 2^(n - 1) L_n = 2^n F_(n + 1) - 1

Theory of Numbers (XIV)

Principle of Mathematical Induction

Fibonacci Number

Lucas number

The Lucas numbers L_n are defined by the equations L_1 = 1 and L_n = F_(n+1) + F_(n-1) for each n > or equal to 2.

Prove that

L_1 + 2L_2 + 4L_3 +8L_4 + ... + 2^(n - 1) L_n = 2^n F_(n + 1) - 1

See the attached file.

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#### Solution Preview

Theory of Numbers (XIV)

Principle of Mathematical Induction

Fibonacci Number

...

#### Solution Summary

This solution is comprised of a detailed explanation of the Lucas numbers L_n . It contains step-by-step explanation of the Lucas numbers L_n defined by the equations L_1 = 1 and

L_n = F_(n+1) + F_(n-1) for each n > 2 and prove of the equation

L_1 + 2L_2 + 4L_3 +8L_4 + ... + 2^(n - 1) L_n = 2^n F_(n + 1) - 1

Solution contains detailed step-by-step explanation. Note is also given at end.