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

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

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{aligned} & -1-\left( -2 \right)=1 \\\ & 0-\left( -1 \right)=1 \\\ & 1-0=1 \\\ & 2-1=1 \\\ \end{aligned}$$ Here, the difference between the terms of this sequence is 1. Hence, it is an AP. The first term of this AP is -2. Common difference is 1. Then the $n^{th}$ term is given by $\begin{aligned} & {{a}_{n}}=-2+\left( n-1 \right)1 \\\ & =-2+n-1 \\\ & {{a}_{n}}=n-3 \end{aligned}$ (c) 4.5, 5, 5.5, 6, ... The difference between the terms is given by $$\begin{aligned} & 5-\left( 4.5 \right)=0.5 \\\ & 5.5-\left( 5 \right)=0.5 \\\ & 6-5.5=0.5 \\\ \end{aligned}$$ Here, the difference between the terms of this sequence is 0.5 Hence, it is an AP. The first term of this AP is 4.5. Common difference is 0.5. Then the $n^{th}$ term is given by $\begin{aligned} & {{a}_{n}}=4.5+\left( n-1 \right)0.5 \\\ & =4.5+0.5n-0.5 \\\ & {{a}_{n}}=0.5n+4 \end{aligned}$ (d) a, a+2b, a+4b, a+8b, a+10b, ... The difference between the terms is given by $$\begin{aligned} & \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{aligned}$$ Here, the differences between the terms of this sequence are 2b, 2b, 4b, 2b. Hence, it is not an AP. (e) ${{1}^{2}},{{5}^{2}},{{7}^{2}},73,\ldots $ The difference between the terms is given by $$\begin{aligned} & {{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{aligned}$$ 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 $\begin{aligned} & {{a}_{n}}=1+\left( n-1 \right)24 \\\ & =1+24n-24 \\\ & {{a}_{n}}=24n-23 \end{aligned}$ **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.