Question

Suppose that${\mathbf{a}}_{\mathbf{n}}\mathbf{=}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{-}\mathbf{5}$for$\mathbf{n}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$find a formula for${\mathbf{a}}_{\mathbf{n}}$

Short answer

Required answer is$a_n=a_0-5n$ for$n=1,2,3,...$

Step-by-step solution

Given 

Given sequence is $a_n=a_{n-1}-5$.

Fordata-custom-editor="chemistry" $\mathrm{n}=1,2,....$

Recurrence formula

We need to find a formula for${\mathrm{a}}_{\mathrm{n}}$

Let us repeatedly use the recurrence relation $a_n$to obtain some formula of${\mathrm{a}}_{\mathrm{n}}$ in terms of$a_0$

$$\begin{gathered} a_n=a_{n-1}-5 \\ =a_{n-1}-1.5 \\ =\left(a_{n-2}-5\right)-5 \\ =a_{n-2}-2.5 \end{gathered}$$

$$\begin{gathered} a_n=\left(a_{n-3}-5\right)-2\cdot 5 \\ =a_{n-3}-3.5 \\ =\dots \\ =a_{n-(n-1)}-(n-1)\cdot 5 \end{gathered}$$

$$\begin{gathered} a_n=a_1-5(n-1) \\ =\left(a_0-5\right)-5(n-1) \\ =a_0-5n \end{gathered}$$

Hence, the formula is$a_n=a_0-5n$