Question

What is the explicit rule for the sequence -1, -6, -11, -16?

Answer 1

Hint: To find the explicit rule for the sequence -1, -6, -11, -16 , check whether the sequence is an AP or GP. For an AP, there will be a common difference that can be found by subtracting the first term from the second term, or the second term from the third term. For a GP, there will be a common ratio that can be obtained by dividing the second term by the first term or the third term by the second term. If we get the sequence as an AP, the general form will be ${{a}_{n}}=a+\left( n-1 \right)d$ , where ${{a}_{n}}$ is the general term, a is the first term and d is the common difference. If the sequence is a GP, then we will get the general form as ${{a}_{n}}={{a}_{1}}{{r}^{n-1}}$ ,where ${{a}_{n}}$ is the general term, ${{a}_{1}}$ is the first term of the GP and r is the common ratio.

Complete step-by-step answer:
We have to find the explicit rule for the sequence -1, -6, -11, -16. We can see that
 $\begin{align}
  & -6-\left( -1 \right)=-6+1=-5 \\
 & -11-\left( -6 \right)=-11+6=-5 \\
 & -16-\left( -11 \right)=-16+11=-5 \\
\end{align}$
That is the common difference of the sequence is -5.
$\Rightarrow d=-5$
Hence, we can call the given sequence as an Arithmetic Progression (AP). We know that for an AP, the ${{n}^{th}}$ term is given by
${{a}_{n}}=a+\left( n-1 \right)d$
where, a is the first term and d is the common difference. From the given sequence, we will get $a=-1$ and we have found $d=-5$ . Let us substitute these values in the above formula.
$\Rightarrow {{a}_{n}}=-1+\left( n-1 \right)-5$
Let us perform distributive property on the second term.
$\begin{align}
  & \Rightarrow {{a}_{n}}=-1+-5n+5 \\
 & \Rightarrow {{a}_{n}}=-5n+4 \\
\end{align}$
Hence, the explicit rule for the sequence -1, -6, -11, -16 is ${{a}_{n}}=-5n+4$ .

Note: There are mainly two types of sequences, that is, Arithmetic Progression (AP) and Geometric Progression (GP). We have to first check what type of progression is the sequence by finding the common difference or common ratio. If a sequence has a common difference, then that sequence will be an AP. If the sequence has a common ratio, then it will be a GP. We have found for this question that the explicit rule is ${{a}_{n}}=-5n+4$ . Let us verify the answer. For this, we have substituted the values for n.
When $n=1$ , we will get the first term.
$\Rightarrow {{a}_{1}}=-5\times 1+-5+4=-1$
When $n=2$ , we will get the second term.
$\Rightarrow {{a}_{2}}=-5\times 2+4=-10+4=-6$
When $n=3$ , we will get the third term.
$\Rightarrow {{a}_{3}}=-5\times 3+4=-15+4=-11$
When $n=4$ , we will get the fourth term.
$\Rightarrow {{a}_{4}}=-5\times 4+4=-20+4=-16$
Hence, the sequence will be -1, -6, -11, -16. Hence, we verified the answer.