Arithmetic-geometric progression

Arithmetic-geometric progression - a sequence of numbers $u_{n}$ ${\ displaystyle u_ {n}}$ $u _ {{n}}$ defined by the recurrence relation $u_{n+1}=qu_{n}+d$ ${\ displaystyle u_ {n + 1} = qu_ {n} + d}$ $u _ {{n + 1}} = qu _ {{n}} + d$ where $q$ ${\ displaystyle q}$ $q$ and $d$ ${\ displaystyle d}$ $d$ Are constants ^[1] . Particular cases of arithmetic-geometric progression are arithmetic progression (with $q=1$ ${\ displaystyle q = 1}$ $q = 1$ ) and geometric progression (at $d=0$ ${\ displaystyle d = 0}$ $d = 0$ )

Common Member Formula

Consider the initial ratio: $u_{n+1}=qu_{n}+d$ ${\ displaystyle u_ {n + 1} = qu_ {n} + d}$ at $n=1,2,...$ ${\ displaystyle n = 1,2, ...}$

Let in this relation $q\neq 1$ ${\ displaystyle q \ neq 1}$ and $d\neq 0$ ${\ displaystyle d \ neq 0}$ . Adding to both parts the expression ${\frac {d}{q-1}}$ ${\ displaystyle {\ frac {d} {q-1}}}$ we get

u_{n+1}+{\dfrac {d}{q-1}}=q\left(u_{n}+{\dfrac {d}{q-1}}\right)

{\ displaystyle u_ {n + 1} + {\ dfrac {d} {q-1}} = q \ left (u_ {n} + {\ dfrac {d} {q-1}} \ right)}

u_{n}+{\dfrac {d}{q-1}}=q\left(u_{n-1}+{\dfrac {d}{q-1}}\right)

{\ displaystyle u_ {n} + {\ dfrac {d} {q-1}} = q \ left (u_ {n-1} + {\ dfrac {d} {q-1}} \ right)}

\ldots

{\ displaystyle \ ldots}

u_{3}+{\dfrac {d}{q-1}}=q\left(u_{2}+{\dfrac {d}{q-1}}\right)

{\ displaystyle u_ {3} + {\ dfrac {d} {q-1}} = q \ left (u_ {2} + {\ dfrac {d} {q-1}} \ right)}

u_{2}+{\dfrac {d}{q-1}}=q\left(u_{1}+{\dfrac {d}{q-1}}\right)

{\ displaystyle u_ {2} + {\ dfrac {d} {q-1}} = q \ left (u_ {1} + {\ dfrac {d} {q-1}} \ right)}

Multiplying the indicated equalities and reducing the same factors (or substituting the left side of the next order equation instead of the brackets on the right side), we obtain the explicit formula of the arithmetic-geometric progression term:

u_{n+1}=q^{n}(u_{1}+{\frac {d}{q-1}})-{\frac {d}{q-1}}

{\ displaystyle u_ {n + 1} = q ^ {n} (u_ {1} + {\ frac {d} {q-1}}) - {\ frac {d} {q-1}}}

Properties

Arithmetic-geometric progression is a second-order return sequence and is given by the equation:

u_{n+1}=(q+1)u_{n}-qu_{n-1}

{\ displaystyle u_ {n + 1} = (q + 1) u_ {n} -qu_ {n-1}}

Difference $d$ ${\ displaystyle d}$ arithmetic-geometric progression is determined by the formula

d={\frac {u_{n}^{2}-u_{n-1}u_{n+1}}{u_{n}-u_{n-1}}}

{\ displaystyle d = {\ frac {u_ {n} ^ {2} -u_ {n-1} u_ {n + 1}} {u_ {n} -u_ {n-1}}}}

Sequence $a_{n}=u_{n+1}-u_{n}$ ${\ displaystyle a_ {n} = u_ {n + 1} -u_ {n}}$ is a geometric progression with the same denominator $q$ ${\ displaystyle q}$ .

The sequence of partial sums of the members of the arithmetic-geometric progression is a third-order return sequence and is given by the equation:

S_{n+1}=(q+2)S_{n}-(2q+1)S_{n-1}+qS_{n-2}

{\ displaystyle S_ {n + 1} = (q + 2) S_ {n} - (2q + 1) S_ {n-1} + qS_ {n-2}}

If the sequence of partial sums is an arithmetic-geometric progression, then the sequence itself is a geometric progression.

Notes

↑ Cloth ya. N. Arithmetic-geometric progression // Quantum . - 1975. - No. 1. - S. 36—39.

[Kvant-1] Cloth ya. N. Arithmetic-geometric progression // Quantum . - 1975. - No. 1. - S. 36—39.

Arithmetic-geometric progression

Common Member Formula

Properties

Notes

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