Question

Find at least three different sequences beginning with the terms 3, 5, 7 whose terms are generated by a simple formula or rule.

Short answer

Hence, the required three different sequences beginning with the terms 3, 5, 7 whose terms are generated by a simple formula or rule are:

$\begin{array}{r}{\mathrm{x}}_{\mathrm{n}}=2\mathrm{n}+1,\text{where}\mathrm{n}=1,2,3,\dots \dots \\ {\mathrm{x}}_{\mathrm{n}}=3+2(\mathrm{n}-1),\text{where}\mathrm{n}=1,2,3,\dots \dots \\ {\mathrm{x}}_{\mathrm{n}}=2+(2\mathrm{n}-1),\text{where}\mathrm{n}=1,2,3,\dots \end{array}$

Step-by-step solution

Consider the formula 2n + 1 and check if it satisfies the series 3,5,7…

We are given that the sequence should start with the terms 3, 5, 7.

Consider the formula, $x_n=2n+1$

$\begin{array}{r}{x}_1=2\left(1\right)+1\\ =3\\ x_2=2\left(2\right)+1\\ =5\\ x_3=2\left(3\right)+1\\ =7\end{array}$

Therefore the required sequence is:

$x_n=2n+1\text{, where}n=1,2,3,\dots \dots$

Consider the formula 3+2 (n - 1) and check if it satisfies the series 3,5,7…

We are given that the sequence should start with the terms 3, 5, 7.

Consider the formula, ${\mathrm{x}}_{\mathrm{n}}=3+2(\mathrm{n}-1)$

role="math" localid="1668662230605" $$\begin{gathered} \begin{array}{r}{\mathrm{x}}_1=3+2(1-1)\\ =3\\ {\mathrm{x}}_2=3+2(2-1)\\ =3+2\left(1\right)\\ =5\end{array} \\ \begin{array}{r}{x}_3=3+2(3-1)\\ =3+2\left(2\right)\\ =7\end{array} \end{gathered}$$

Therefore the required sequence is:

$x_n=3+2(n-1),\text{where}n=1,2,3,\dots \dots$

Consider the formula 2+ (2n - 1) and check if it satisfies the series 3,5,7…

We are given that the sequence should start with the terms 3, 5, 7.

Consider the formula, $x_n=2+\left(2n-1\right)$.

$\begin{array}{r}{\mathrm{x}}_1=2+\left(2\right(1)-1)\\ =2+1\\ =3\\ {\mathrm{x}}_2=2+\left(2\right(2)-1)\\ =2+3\\ =5\\ {\mathrm{x}}_3=2+\left(2\right(3)-1)\\ =2+5\\ =7\end{array}$

Therefore the required sequence is:

$x_n=2+(2n-1),\text{where}n=1,2,3,\dots \dots$