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?

Guest Sep 6, 2021

#1**+1 **

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.)

Alan Sep 6, 2021

#2**0 **

*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 100 ^{th} 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

Guest Sep 6, 2021

edited by
Guest
Sep 6, 2021