Question

Give a recursive definition of the sequence\(\left\{ {{{\bf{a}}_{\bf{n}}}} \right\}\),\({\bf{n = 1,2,3,}}...\)if

a)\({{\bf{a}}_{\bf{n}}}{\bf{ = 6n}}\)b)\({{\bf{a}}_{\bf{n}}}{\bf{ = 2n + 1}}\)

c)\({{\bf{a}}_{\bf{n}}}{\bf{ = 1}}{{\bf{0}}^{\bf{n}}}\)d)\({{\bf{a}}_{\bf{n}}}{\bf{ = 5}}\).

Short answer

1. The recursive definition of the sequence is\({a_{n + 1}} = {a_n} + 6\)for\(n \ge 1\)and\({a_1} = 6\).
2. Therecursive definition of the sequence is\({a_{n + 1}} = {a_n} + 2\)for\(n \ge 1\)and\({a_1} = 3\).
3. The recursive definition of the sequence is\({a_{n + 1}} = 10{a_n}\)for\(n \ge 1\)and\({a_1} = 10\).
4. The recursive definition of the sequence is \({a_{n + 1}} = {a_n}\) for \(n \ge 1\) and \({a_1} = 5\).

Step-by-step solution

Identification of the given data

The given data can be listed below as,

• The value of \(n\) is, \(n = 1,2,3,...\).

Significance of sequences

A sequence can also be defined recursively, meaning that the previous terms define successive terms in the sequence. The recursive sequence is obtained by the deriving each successive term such that it is 2 larger than the previous obtained term.

(a) Determination of the recursive definition of the sequence\({{\bf{a}}_{\bf{n}}}{\bf{ = 6n}}\)

The given expression of the sequence is represented as,

\({a_n} = 6n\) ….. (1)

For\(n = 1\), the equation (1) can be represented as,

\(\begin{aligned}{c}{a_1} = 6\left( 1 \right)\\ = 6\end{aligned}\)

For\(n = 2\), the equation (1) can be represented as,

\(\begin{aligned}{c}{a_2} = 6\left( 2 \right)\\ = 12\end{aligned}\)

For\(n = 3\), the equation (1) can be represented as,

\(\begin{aligned}{c}{a_3} = 6\left( 3 \right)\\ = 18\end{aligned}\)

Substitute\(\left( {n + 1} \right)\)for\(n\)in the above equation (1).

\(\begin{aligned}{c}{a_{n + 1}} = 6\left( {n + 1} \right)\\ = 6n + 6\end{aligned}\) …. (2)

Substitute the values in the equation (2).

\({a_{n + 1}} = {a_n} + 6\)

Thus, the recursive definition of the given sequence is \({a_{n + 1}} = {a_n} + 6\) for \(n \ge 1\) and\({a_1} = 6\).

(b) Determination of the recursive definition of the sequence\({{\bf{a}}_{\bf{n}}}{\bf{ = 2n + 1}}\)

The given expression of the sequence is represented as,

\({a_n} = 2n + 1\) …… (2)

For\(n = 1\), the equation (3) can be represented as,

\(\begin{aligned}{c}{a_1} = 2\left( 1 \right) + 1\\ = 3\end{aligned}\)

For\(n = 2\), the equation (3) can be represented as,

\(\begin{aligned}{c}{a_2} = 2\left( 2 \right) + 1\\ = 5\end{aligned}\)

For\(n = 3\), the equation (3) can be represented as,

\(\begin{aligned}{c}{a_3} = 2\left( 3 \right) + 1\\ = 7\end{aligned}\)

Substitute\(\left( {n + 1} \right)\)for\(n\)in the above equation (3).

\(\begin{aligned}{c}{a_{n + 1}} = 2\left( {n + 1} \right) + 1\\ = \left( {2n + 1} \right) + 2\end{aligned}\) …… (4)

Substitute the values in the equation (4).

\({a_{n + 1}} = {a_n} + 2\)

Thus, the recursive definition of the given sequence is \({a_{n + 1}} = {a_n} + 2\) for \(n \ge 1\) and \({a_1} = 3\).

(c) Determination of the recursive definition of the sequence\({{\bf{a}}_{\bf{n}}}{\bf{ = 1}}{{\bf{0}}^{\bf{n}}}\)

The given expression of the sequence is represented as,

\({a_n} = {10^n}\) …… (5)

For\(n = 1\), the equation (5) can be represented as,

\(\begin{aligned}{c}{a_1} = {10^1}\\ = 10\end{aligned}\)

For\(n = 2\), the equation (5) can be represented as,

\(\begin{aligned}{c}{a_2} = {10^2}\\ = 100\end{aligned}\)

For\(n = 3\), the equation (5) can be represented as,

\(\begin{aligned}{c}{a_3} = {10^3}\\ = 1000\end{aligned}\)

Substitute\(\left( {n + 1} \right)\)for\(n\)in the above equation (5).

\(\begin{aligned}{c}{a_{n + 1}} = {10^{\left( {n + 1} \right)}}\\ = \left( {{{10}^n}} \right){\left( {10} \right)^1}\\ = 10{\left( {10} \right)^n}\end{aligned}\) …… (6)

Substitute the values in the equation (6).

\({a_{n + 1}} = 10{a_n}\)

Thus, the recursive definition of the given sequence is \({a_{n + 1}} = 10{a_n}\) for \(n \ge 1\) and \({a_1} = 10\).

(d) Determination of the recursive definition of the sequence\({{\bf{a}}_{\bf{n}}}{\bf{ = 5}}\)

The given expression of the sequence is represented as,

\({a_n} = 5\) …… (7)

For\(n = 1\), the equation (7) can be represented as,

\({a_1} = 5\)

For\(n = 2\), the equation (7) can be represented as,

\({a_2} = 5\)

For\(n = 3\), the equation (7) can be represented as,

\({a_3} = 5\)

Substitute\(\left( {n + 1} \right)\)for\(n\)in the above equation (7).

\({a_{n + 1}} = 5\)

Substitute the values in the equation (8).

\({a_{n + 1}} = {a_n}\)

Thus, the recursive definition of the given sequence is \({a_{n + 1}} = {a_n}\) for \(n \ge 1\) and \({a_1} = 5\).