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# Sequence

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The Lucas sequence is the sequence 1, 3, 4, 7, 11,... where the first term is 1, the second term is 3 and each term after that is the sum of the previous two terms. What is the remainder when the 100th term of the sequence is divided by 20?

Sep 6, 2021

#1
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A short computer program gives the 100'th term as 792070839848372253127, so the remainder on dividing by 20 is 7.

(An internet search might well produce a closed form expression for the n'th term.)

Sep 6, 2021
#2
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An internet search might well produce a closed form expression for the n'th term.

You’re right; there is one:

\(L_n = \dfrac{1-sqrt(5)}{2}^n + \; \dfrac{1+sqrt(5)}{2}^n \)

The sequence starts with n=0, so the 100th term is n=99

n=99;((1+sqrt(5))/2)^n+((1-sqrt(5))/2)^n

The web2.0calc has a rounding error greater than -2.6

Wolfram gives (489526700523968661124). Dividing this by (20) gives a remainder of (4).

GA

Sep 6, 2021
edited by Guest  Sep 6, 2021