Question

Determine whether each of the following sequences is an AP or not. If it is AP, then find its $n^{th}$ term.
(a) 111, 107, 103, 9, ...
(b) -2, -1, 0, 1, 2, ...
(c) 4.5, 5, 5.5, 6, ...
(d) a, a+2b, a+4b, a+8b, a+10b, ...
(e) ${{1}^{2}},{{5}^{2}},{{7}^{2}},73,\ldots $

Answer 1

Hint: First we will find the difference between the terms of a sequence. If it is the same for all terms, then it is an AP. Once, we know a sequence is a AP or not, then we will find the first term, common difference, and then its $n^{th}$ term with the help of the formula ${{a}_{n}}=a+\left( n-1 \right)d$.

Complete step by step answer:
We know, in a AP with the first term a, common difference d, number of terms n, the $n^{th}$ term ${{a}_{n}}$ is given by
${{a}_{n}}=a+\left( n-1 \right)d$

(a) 111, 107, 103, 9, ...
The difference between the terms is given by
\[\begin{array}{*{35}{l}}
   107-111=-4 \\
   103-107=-4 \\
   9-103=-94 \\
\end{array}\]
Here, the differences between the terms of this sequence are -4, -4, -94.
Hence, it is not an AP.

(b) -2, -1, 0, 1, 2, ...
The difference between the terms is given by
\[\begin{align}
  & -1-\left( -2 \right)=1 \\
 & 0-\left( -1 \right)=1 \\
 & 1-0=1 \\
 & 2-1=1 \\
\end{align}\]
Here, the difference between the terms of this sequence is 1.
Hence, it is an AP.
The first term of this AP is -2. Common difference is 1.
Then the $n^{th}$ term is given by
 $\begin{align}
  & {{a}_{n}}=-2+\left( n-1 \right)1 \\
 & =-2+n-1 \\
 & {{a}_{n}}=n-3
\end{align}$

(c) 4.5, 5, 5.5, 6, ...
The difference between the terms is given by
\[\begin{align}
  & 5-\left( 4.5 \right)=0.5 \\
 & 5.5-\left( 5 \right)=0.5 \\
 & 6-5.5=0.5 \\
\end{align}\]
Here, the difference between the terms of this sequence is 0.5
Hence, it is an AP.
The first term of this AP is 4.5. Common difference is 0.5.
Then the $n^{th}$ term is given by
 $\begin{align}
  & {{a}_{n}}=4.5+\left( n-1 \right)0.5 \\
 & =4.5+0.5n-0.5 \\
 & {{a}_{n}}=0.5n+4
\end{align}$

(d) a, a+2b, a+4b, a+8b, a+10b, ...
The difference between the terms is given by
\[\begin{align}
  & \begin{array}{*{35}{l}}
   a+2b-\left( a \right)=2b \\
   a+4b-\left( a+2b \right)=2b \\
   a+8b-\left( a+4b \right)=4b \\
\end{array} \\
 & a+10b-\left( a+8b \right)=2b \\
\end{align}\]
Here, the differences between the terms of this sequence are 2b, 2b, 4b, 2b.
Hence, it is not an AP.

(e) ${{1}^{2}},{{5}^{2}},{{7}^{2}},73,\ldots $
The difference between the terms is given by
\[\begin{align}
  & {{5}^{2}}-\left( {{1}^{2}} \right)=25-1=24 \\
 & {{7}^{2}}-\left( {{5}^{2}} \right)=49-25=24 \\
 & 73-\left( {{7}^{2}} \right)=73-49=24 \\
\end{align}\]
Here, the difference between the terms of this sequence is 24
Hence, it is an AP.
The first term of this AP is 1. Common difference is 24.
Then the $n^{th}$ term is given by
 $\begin{align}
  & {{a}_{n}}=1+\left( n-1 \right)24 \\
 & =1+24n-24 \\
 & {{a}_{n}}=24n-23
\end{align}$

Hence, sequences (a) and (d) are not AP. Sequences (b) is a AP with $n^{th}$ term ${{a}_{n}}=n-3$, (c) is a AP with $n^{th}$ term ${{a}_{n}}=0.5n+4$, (e) is a AP with $n^{th}$ term ${{a}_{n}}=24n-23$.

Note: The best way to deal with such questions is to tackle each sequence individually. An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to each preceding term. This fixed number is called the common difference of an AP.