A recurrence relation is an equation that recursively defines a sequence of values. Each term of the

Question

A recurrence relation is an equation that recursively defines a sequence of values. Each term of the sequence is defined as a function of the preceding terms. Examples Recurrence RelationsDescription of getting the Next TermFirst $5$ Terms ${{x}_{n+1}}=\sqrt{{{x}_{n}}+5}$, ${{x}_{0}}=1$ ${{u}_{n+1}}=0.5{{u}_{n}}+8$, ${{u}_{1}}=4$ ${{a}_{n}}={{a}_{n-1}}+k$, ${{a}_{0}}=3$ ${{u}_{n}}=-2{{u}_{n-1}}$ , ${{u}_{0}}=3$ ${{T}_{n}}={{T}_{n-1}}+{{T}_{n-2}}$, ${{T}_{1}}=0$, ${{T}_{2}}=1$

Accepted Answer

(1) Add $5$, then take the square root: $1,\ 2.4495,\ 2.7294,\ 2.7802,\ 2.7893$, increasing (2) Multiply by $0.5$, then add $8$: $4,\ 10,\ 13,\ 14.5,\ 15.25$, increasing (3) Add $k$: $3,\ 3+k,\ 3+2k,\ 3+3k,\ 3+4k$, increasing for $k>0$, constant for $k=0$, decreasing for $k<0$ (4) Multiply by $-2$: $3,\ -6,\ 12,\ -24,\ 48$, alternating (5) Add the preceding two terms: $0,\ 1,\ 1,\ 2,\ 3$, non-decreasing

Recurrence relation	How the next term is obtained	First terms
\(x_{n+1}=\sqrt{x_n+5},\ x_0=1\)	Add \(5\), then take the positive square root.	\(1,\sqrt6,\sqrt{\sqrt6+5},\ldots\)
\(u_{n+1}=0.5u_n+8,\ u_1=4\)	Halve the current term, then add \(8\).	\(4,10,13,14.5,15.25,\ldots\)
\(a_n=a_{n-1}+k,\ a_0=3\)	Add the constant \(k\).	\(3,3+k,3+2k,3+3k,\ldots\)
\(u_n=-2u_{n-1},\ u_0=3\)	Multiply the current term by \(-2\).	\(3,-6,12,-24,48,\ldots\)
\(T_n=T_{n-1}+T_{n-2},\ T_1=0,\ T_2=1\)	Add the preceding two terms.	\(0,1,1,2,3,\ldots\)

Recurrence Relations

Recurrence Relations

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