Question

Given the recursive formula for an arithmetic sequence, find the explicit formula.

$$\begin{gathered} {\mathrm{a}}_{\mathrm{n}}={\mathrm{a}}_{\mathrm{n}-1}-9 \\ {\mathrm{a}}_1=-35 \end{gathered}$$

• A. ${\mathrm{a}}_{\mathrm{n}}=-26-9\mathrm{n}$
• B. ${\mathrm{a}}_{\mathrm{n}}=-34-7\mathrm{n}$
• C. ${\mathrm{a}}_{\mathrm{n}}=-27-7\mathrm{n}$
• D. ${\mathrm{a}}_{\mathrm{n}}=-28-7\mathrm{n}$

Answer 1

The correct option is A

${\mathrm{a}}_{\mathrm{n}}=-26-9\mathrm{n}$

Explanation for the correct answer:

Option (A):

Given

$$\begin{gathered} {\mathrm{a}}_{\mathrm{n}}={\mathrm{a}}_{\mathrm{n}-1}-9 \\ {\mathrm{a}}_1=-35 \end{gathered}$$

The recursive formula is the function of the previous term (${\mathrm{a}}_{\mathrm{n}-1}$)

$a_n$is the final term

${\mathrm{a}}_1$ is the first term

$\mathrm{d}$ is the common difference.

Finding the explicit formula in terms of $\mathrm{n}$:

The explicit formula is written in the form ${\mathrm{a}}_{\mathrm{n}}={\mathrm{a}}_1+(\mathrm{n}-1)\mathrm{d}$

where, $a_1=-35,\mathrm{d}=-9$

Putting up the values accordingly,

$$\begin{gathered} {\mathrm{a}}_{\mathrm{n}}=-35+(\mathrm{n}-1)(-9) \\ {\mathrm{a}}_{\mathrm{n}}=-35+(-9\mathrm{n}+9) \\ {\mathrm{a}}_{\mathrm{n}}=-35-9\mathrm{n}+9 \\ {\mathrm{a}}_{\mathrm{n}}=-26-9\mathrm{n} \end{gathered}$$

This is the explicit formula, which matches with option $\left(A\right)$.

Hence, the Correct option is (A).

Explanation of the incorrect options:

As the above calculations give the explicit formula as ${\mathrm{a}}_{\mathrm{n}}=-26-9\mathrm{n}$, which does not match with options $\left(B\right),\left(C\right)$ and $\left(D\right)$.

Hence, incorrect options are $\left(B\right),\left(C\right)$ and $\left(D\right)$.

Therefore, option (A) is the correct answer.