Question

Find these terms of the sequence, where,$\left\{a_n\right\}\mathbf{\text{, where}}{\mathit{a}}_{\mathbf{n}}\mathbf{=}\mathbf{2}\mathbf{\cdot }\mathbf{(}\mathbf{-}\mathbf{3}{\mathbf{)}}^{\mathbf{n}}\mathbf{+}{\mathbf{5}}^{\mathbf{n}}$

$$\begin{gathered} \mathbf{a}\mathbf{.}\mathbf{}{\mathbf{a}}_{\mathbf{0}} \\ \mathbf{b}\mathbf{.}\mathbf{}{\mathbf{a}}_{\mathbf{1}} \\ \mathbf{c}\mathbf{.}\mathbf{}{\mathbf{a}}_{\mathbf{4}} \\ \mathbf{d}\mathbf{.}\mathbf{}{\mathbf{a}}_{\mathbf{5}} \end{gathered}$$

Short answer

Hence, the terms of the given sequence are:

$$\begin{gathered} {\mathrm{a}}_0=3 \\ {\mathrm{a}}_1=-1- \\ {\mathrm{a}}_4=787 \\ {\mathrm{a}}_5=2639 \end{gathered}$$

Step-by-step solution

(a)Step 1: Solve for value of n as 0

We find by substituting n = 0 in the given sequence $a_n=2\cdot (-3{)}^n+5^n$ .

$\begin{array}{r}{a}_0=2\cdot (-3{)}^0+5^0\\ =2\cdot 1+1\\ =2+1\\ =3\end{array}$

(b)Step 2: Solve for value of n as 1

We find $a_1$by substituting n = 1in the given sequence $a_n=2\cdot (-3{)}^n+5^n$.

$\begin{array}{r}{a}_1=2\cdot (-3{)}^1+5^1\\ =2\cdot (-3)+5\\ =-6+5\\ =-1\end{array}$

(c)Step 3: Solve for value of n as 4

We find $a_4$by substituting n = 4 in the given sequence $a_n=2\cdot (-3{)}^n+5^n$.

$\begin{array}{r}{a}_4=2\cdot (-3{)}^4+5^4\\ =2\cdot \left(81\right)+625\\ =162+625\\ =787\end{array}$

(d)Step 4: Solve for value of n as 5

We find$a_5$by substituting n = 5in the given sequence$a_n=2\cdot (-3{)}^n+5^n$

$\begin{array}{r}{a}_5=2\cdot (-3{)}^5+5^5\\ =2\cdot (-243)+3125\\ =-486+3125\\ =2639\end{array}$