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MATH / The infinite sequence T={t₀,t₁,t₂,…} is defined as t₀=0, t₁=1, and t_(n)=t_(n−2)+t_(n−1) for all integers n>1. If a, b, c are fixed non-negative integers such that ……

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problem

The infinite sequence T={t0,t1,t2,…}T=\{t_0,t_1,t_2,\ldots\} is defined as t0=0,t_0=0, t1=1,t_1=1, and tn=tn−2+tn−1t_n=t_{n-2}+t_{n-1} for all integers n>1.n>1. If a,a, b,b, cc 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 ta+tb+tct_a+t_b+t_c is divided by $7?$
Plain-text mathematical notation (without MathML)
The infinite sequence T={t₀,t₁,t₂,…} is defined as t₀=0, t₁=1, and t_(n)=t_(n−2)+t_(n−1) for all integers n>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 $7?$
Original LaTeX notation
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 $n>1.$ 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 $7?$

level

Level 5

type

Number Theory

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Source and history

task_id
ad4c0299-0cb0-5594-bf49-2662f4c9e8c9
task_key
number~5ftheory--test--ad4c0299-0cb0-5594-bf49-2662f4c9e8c9
task_revision_id
1
config / split
number_theory / test

Official source

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