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?
+33661 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.)
+2489 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
