Answer to Question #192820 in Discrete Mathematics for SUMIT KUMAR SINGH

Question

a_n = a _0.5n + n, a₁ = 0, where n is a power of 2, is a linear recurrence relation. ( true / falsa ).

Answer 1

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 $u_n=\varphi (n,u_{n-1},u_{n-2},\ldots ,u_{n-d}),$ where ${\displaystyle \varphi :\mathbb {N} \times X^{d}\to X} , \ \varphi (n,u_{n-1},u_{n-2},\ldots ,u_{n-d})=c_1u_{n-1}+c_2u_{n-2}+\ldots +c_da_{n-d}+f(n),$

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

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

Answer: false