# Sequences math problems and answers Recent questions in Sequences
Sequences
ANSWERED ### Show that the sequence $$\displaystyle{a}_{{{n}}}$$ is an solution of the recurrence relation $$\displaystyle{a}_{{{n}}}=-{3}{a}_{{{n}-{1}}}+{4}{a}_{{{n}-{2}}}\ {\quad\text{if}\quad}\ {a}_{{{n}}}={\left(-{4}\right)}^{{{n}}}$$.

Sequences
ANSWERED ### Find a recurrence relation satisfied by this sequence. $$\displaystyle{a}_{{{n}}}={2}{n}+{3}$$

Sequences
ANSWERED ### Show that the sequence $$\displaystyle{a}_{{{n}}}$$ is an solution of the recurrence relation $$\displaystyle{a}_{{{n}}}=-{3}{a}_{{{n}-{1}}}+{4}{a}_{{{n}-{2}}}\ {\quad\text{if}\quad}\ {a}_{{{n}}}={2}{\left(-{4}\right)}^{{{n}}}+{3}$$.

Sequences
ANSWERED ### Write a formula for the nth term of the arithmetic sequence 15, 20, 25, 30, ...

Sequences
ANSWERED ### Find a recurrence relation satisfied by this sequence. $$\displaystyle{a}_{{{n}}}={n}^{{{2}}}+{n}$$

Sequences
ANSWERED ### Show that the sequence $$\displaystyle{a}_{{{n}}}$$ is an solution of the recurrence relation $$\displaystyle{a}_{{{n}}}=-{3}{a}_{{{n}-{1}}}+{4}{a}_{{{n}-{2}}}\ {\quad\text{if}\quad}\ {a}_{{{n}}}={0}$$.

Sequences
ANSWERED ### Substitute n=1, 2, 3, 4, 5 and find the first five sequences in sequence $$\left\{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dotsm+\frac{1}{2^{n}}\right\}$$

Sequences
ANSWERED ### Find a recurrence relation satisfied by this sequence. $$\displaystyle{a}_{{{n}}}={n}^{{{2}}}$$

Sequences
ANSWERED ### Find a recurrence relation satisfied by this sequence. $$\displaystyle{a}_{{{n}}}={2}{n}$$

Sequences
ANSWERED ### Show that the sequence $$\displaystyle{a}_{{{n}}}$$ is an solution of the recurrence relation $$\displaystyle{a}_{{{n}}}=-{3}{a}_{{{n}-{1}}}+{4}{a}_{{{n}-{2}}}\ {\quad\text{if}\quad}\ {a}_{{{n}}}={1}$$.

Sequences
ANSWERED ### Give the first six terms of the following sequences. You can assume that the sequences start with an index of 1. 1) An arithmetic sequence in which the first value is 2 and the common difference is 3. 2) A geometric sequence in which the first value is 27 and the common ratio is $$\displaystyle{\frac{{{1}}}{{{3}}}}$$

Sequences
ANSWERED ### Find a recurrence relation satisfied by this sequence. $$\displaystyle{a}_{{{n}}}={n}+{\left(-{1}\right)}^{{{n}}}$$

Sequences
ANSWERED ### Prove that following sequence is Cauchy, using just the definition $$\displaystyle\le{f}{t}{\left\lbrace{\frac{{{\left(-{1}\right)}^{{{n}}}}}{{{n}}}}{r}{i}{g}{h}{t}\right\rbrace}^{{\infty}}_{\left\lbrace{n}={1}\right\rbrace}$$

Sequences
ANSWERED ### Find a recurrence relation satisfied by this sequence. $$\displaystyle{a}_{{{n}}}={3}$$

Sequences
ANSWERED ### Find a recurrence relation satisfied by this sequence. $$\displaystyle{a}_{{{n}}}={5}^{{{n}}}$$

Sequences
ANSWERED ### Find a recurrence relation satisfied by this sequence. $$\displaystyle{a}_{{{n}}}={n}!$$

Sequences
ANSWERED ### Do the first 6 terms of the sequence $$\displaystyle{g}_{{1}}={2}$$ $$\text{and} \ g_2=1$$. The rest of the terms are given by the formula $$\displaystyle{g}_{{n}}={n}{g}_{{{n}-{1}}}+{g}_{{{n}-{2}}}$$

Sequences
ANSWERED ### Find the ﬁrst four terms of each of the recursively deﬁned sequences $$\displaystyle{d}_{{k}}={k}{\left({d}_{{{k}-{1}}}\right)}^{{{2}}}$$, for all integers $$\displaystyle{k}\geq{1}$$ $$\displaystyle{d}_{{0}}={3}$$

Sequences
ANSWERED ### Find the limits of the following sequences $$\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\frac{{{n}\cdot{\ln{{\left({n}\right)}}}}}{{{n}^{{{2}}}+{1}}}}$$ $$\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\frac{{{\sin{{\left({n}\right)}}}}}{{{n}}}}$$
ANSWERED 