Question

What terms are ${\mathbf{a}}_{\mathbf{0}}\mathbf{,}{\mathbf{a}}_{\mathbf{1}}\mathbf{,}{\mathbf{a}}_{\mathbf{2}}\mathbf{\text{and}}{\mathbf{a}}_{\mathbf{3}}$of the sequence $\left\{a_n\right\}$if ${\mathbf{a}}_{\mathbf{n}}$equals

$\begin{array}{r}{\mathbf{2}}^{\mathbf{n}}\mathbf{+}\mathbf{1}\mathbf{?}\\ \mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}{\mathbf{)}}^{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{?}\\ \mathbf{[}\mathbf{n}\mathbf{/}\mathbf{2}\mathbf{]}\mathbf{?}\\ \mathbf{[}\mathbf{n}\mathbf{/}\mathbf{2}\mathbf{]}\mathbf{+}\mathbf{[}\mathbf{n}\mathbf{/}\mathbf{2}\mathbf{]}\mathbf{?}\end{array}$

Short answer

Hence, the terms of the given sequence are:

(a)For the sequence $2^{\mathrm{n}}+1,a_0=2,a_1=3,a_2=5,a_3=9$

(b) For the sequence role="math" $(n+1{)}^{n+1},a_0=1,a_1=4,a_2=27,a_3=256$

(c) For the sequence $[n/2],a_0=0,a_1=1/2,a_2=1,a_3=3/2$

(d) For the sequence$[n/2]+[n/2],a_0=0,a_1=1,a_2=2,a_3=3$

Step-by-step solution

(a)Step 1: Solve for the different values of n in the sequence

We find ${\mathrm{a}}_0,{\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3$by substituting n = 0,1,2,3in the given sequence $2^n+1$

$\begin{array}{r}{\mathrm{a}}_0=2^0+1\\ =1+1\\ =2\\ a_1=2^1+1\\ =2+1\\ =3\\ a_2=2^2+1\\ =4+1\\ =5\\ a_3=2^3+1\\ =8+1\\ =9\end{array}$

(b)Step 2: Solve for the different values of n in the sequence (n+1)n+1

We find ${\mathrm{a}}_0,{\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3$by substituting n = 0,1,2,3 in the given sequence ${\left(\mathrm{n}+1\right)}^{\mathrm{n}+1}$.

$\begin{array}{r}{\mathrm{a}}_0=(0+1{)}^{0+1}\\ =(1{)}^1\\ =1\\ {\mathrm{a}}_1=(1+1{)}^{1+1}\\ =(2{)}^2\\ =4\\ {\mathrm{a}}_2=(2+1{)}^{2+1}\\ =(3{)}^3\\ =27\\ {\mathrm{a}}_3=(3+1{)}^{3+1}\\ =(4{)}^4\\ =256\end{array}$

(c)Step 3: Solve for the different values of n in the sequence [[n/2]

We find ${\mathrm{a}}_0,{\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3$by substituting n = 0,1,2,3in the given sequence [n/2].

$\begin{array}{r}{\mathrm{a}}_0=[0/2]\\ =0\\ {\mathrm{a}}_1=[1/2]\\ =1/2\\ {\mathrm{a}}_2=[2/2]\\ =1\\ {\mathrm{a}}_3=[3/2]\\ =3/2\end{array}$

(d)Step 4: Solve for the different values of n in the sequence [n/2] + [n/2]

We find ${\mathrm{a}}_0,{\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3$by substituting n = 0,1,2,3 in the given sequence [n/2] + [n/2].

$$\begin{gathered} \begin{array}{r}{\mathrm{a}}_0=[0/2]+[0/2]\\ =0+0\\ =0\end{array} \\ \begin{array}{r}{\mathrm{a}}_1=[1/2]+[1/2]\\ =1/2+1/2\\ =1\\ {\mathrm{a}}_2=[2/2]+[2/2]\\ =1+1\\ =2\\ {\mathrm{a}}_3=[3/2]+[3/2]\\ =3/2+3/2\\ =3\end{array} \end{gathered}$$