GATE CS 2023 Set 1 — Question 15
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Engineering Mathematics → Sets & Combinatorics → Recurrence Relations
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Question
The Lucas sequence is defined by the recurrence relation:with and .
Which one of the options given is TRUE?
Which one of the options given is TRUE?
Correct answer
(A) Lₙ = (1 + √(5)2)ⁿ + (1 - √(5)2)ⁿ
Solution
The characteristic equation for the recurrence is . The roots are and .
The general solution is .
Using initial conditions:
For :
For :
Since and , we have:
.
Substituting into the equation:
.
Then .
Thus, .
The general solution is .
Using initial conditions:
For :
For :
Since and , we have:
.
Substituting into the equation:
.
Then .
Thus, .
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