Question #192820

an = a 0.5n + n, a1 = 0, where n is a power of 2, is a linear recurrence relation. ( true / falsa ).


1
Expert's answer
2021-05-17T13:50:02-0400

A linear recurrence relation is an equation that expresses each element of a sequence as a linear function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a linear recurrence relation has the form un=φ(n,un1,un2,,und),u_n=\varphi (n,u_{n-1},u_{n-2},\ldots ,u_{n-d}), where φ:N×XdX, φ(n,un1,un2,,und)=c1un1+c2un2++cdand+f(n),{\displaystyle \varphi :\mathbb {N} \times X^{d}\to X} , \ \varphi (n,u_{n-1},u_{n-2},\ldots ,u_{n-d})=c_1u_{n-1}+c_2u_{n-2}+\ldots +c_da_{n-d}+f(n),

is a everywhere defined function that involves dd  consecutive elements of the sequence. In this case, dd  initial values are needed for defining a sequence.


Taking into account that an=a0.5n+na_n = a_{0.5n} + n is defined only for nn equals to the powers of 2, and hence is not defined for the rest natural numbers, the previous definition implies that an=a0.5n+na_n = a_{0.5n} + n is not a linear recurrence relation.


Answer: false


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