Question

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

$$\begin{gathered} \mathbf{(}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{n}}\mathbf{?} \\ \mathbf{3}\mathbf{}\mathbf{?} \\ \mathbf{7}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{4}}^{\mathbf{n}}\mathbf{?} \\ {\mathbf{2}}^{\mathbf{n}}\mathbf{+}{\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}}^{\mathbf{n}}\mathbf{?} \end{gathered}$$

Short answer

Hence, the terms of the given sequence are:

(a)For the sequence$(-2{)}^n,a_0=1,a_1=-2,a_2=4,a_3=-8$.

(b) For the sequence$3,a_0=3,a_1=3,a_2=3,a_3=3$

(c) For the sequence$7+4^n,a_0=8,a_1=11,a_2=23,a_3=71$

(d) For the sequence$2^{\mathrm{n}}+(-2{)}^{\mathrm{n}},{\mathrm{a}}_0=2,{\mathrm{a}}_1=0,{\mathrm{a}}_2=8,{\mathrm{a}}_3=0$

Step-by-step solution

(a)Step 1: Solve for the different values of n in the sequence (-2)n

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

$$\begin{gathered} (-2{)}^{\mathrm{n}} \\ \begin{array}{r}{\mathrm{a}}_0=(-2{)}^0\\ =1\\ {\mathrm{a}}_1=(-2{)}^1\\ =-2\\ {\mathrm{a}}_2=(-2{)}^2\\ =4\\ {\mathrm{a}}_3=(-2{)}^3\\ =-8\end{array} \end{gathered}$$

(b)Step 2: Solve for the different values of n in the sequence 3

The given sequence 3 is a single term sequence and hence,

$\begin{array}{r}{a}_0=3\\ a_1=3\\ a_2=3\\ a_3=3\end{array}$

(c)Step 3: Solve for the different values of n in the sequence 7+4n

We find ${\mathrm{a}}_0,{\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3$by substituting n = 0,1,2,3respectively in the given sequence $7+4^n$

$\begin{array}{r}{\mathrm{a}}_0=7+4^0\\ =7+1\\ =8\\ {\mathrm{a}}_1=7+4^1\\ =7+4\\ =11\\ {\mathrm{a}}_2=7+4^2\\ =7+16\\ =23\\ {\mathrm{a}}_3=7+4^3\\ =7+64\\ =71\end{array}$

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

We find $a_0,a_1,a_2,a_3$by substituting n = 0,1,2,3respectively in the given sequence$2^n+\left(-2^n\right)$

$\begin{array}{r}{a}_0=2^0+(-2{)}^0\\ =1+1\\ =2\\ a_1=2^1+(-2{)}^1\\ =2-2\\ =0\end{array}$

$\begin{array}{r}{a}_2=2^2+(-2{)}^2\\ =4+4\\ =8\\ a_3=2^3+(-2{)}^3\\ =8-8\\ =0\end{array}$