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labwork [276]
3 years ago
12

The infinite sequence $T=\{t_0,t_1,t_2,\ldots\}$ is defined as $t_0=0,$ $t_1=1,$ and $t_n=t_{n-2}+t_{n-1}$ for all integers $n&g

t;1.$ If $a,$ $b,$ $c$ are fixed non-negative integers such that\begin{align*} a&\equiv 5\pmod {16}\\ b&\equiv 10\pmod {16}\\ c&\equiv 15\pmod {16}, \end{align*}then what is the remainder when $t_a+t_b+t_c$ is divided by $7?$ You can use a LaTeX renderer to see what this says.
Mathematics
1 answer:
katrin [286]3 years ago
6 0

t_n are the Fibonacci numbers,

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

Taken modulo 7, we get the sequence

0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, ...

that repeats with period 16 (this is known as the 7th Pisano period).

This is to say,

t_a\equiv t_{16n+5}\equiv t_5\equiv5\pmod7

t_b\equiv t_{16n+10}\equiv t_{10}\equiv6\pmod7

t_c\equiv t_{16n+15}\equiv t_{15}\equiv1\pmod7

so that

(t_a+t_b+t_c)\equiv(5+6+1)\equiv\boxed5\pmod7

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