Question

For the following AP’s, write the first term and the common difference:
(1) $3,1, - 1, - 3,....$
(2) $ - 5, - 1,3,7,....$
(3) $\dfrac{1}{3},\dfrac{5}{3},\dfrac{9}{3},\dfrac{{13}}{3}....$
(4) $0.6,1.7,2.8,3.9,....$

Answer 1

Hint:A sequence of numbers is called an Arithmetic progression if the difference between any two consecutive terms is always the same.From the given sequence write the first term and find the difference between two consecutive numbers which gives a common difference.

Complete step-by-step answer:
An arithmetic progression is given by $a,\left( {a + d} \right),\left( {a + 2d} \right),\left( {a + 3d} \right),......$
Where $a$ is the first term of AP and $d$ is the common difference of AP.
(1) $3,1, - 1, - 3,....$
Here, first term, $a = 3$
Common difference, $d = $\[second{\text{ }}term - first{\text{ }}term\]
Or $d = {a_2} - {a_1}$
$ \Rightarrow d = 1 - 3$
$ \Rightarrow d = - 2$

(2) $ - 5, - 1,3,7,....$
Here, first term, $a = - 5$
Common difference, $d = $\[second{\text{ }}term - first{\text{ }}term\]
Or $d = {a_2} - {a_1}$
$ \Rightarrow d = - 1 - \left( { - 5} \right)$
$ \Rightarrow d = - 1 + 5$
$ \Rightarrow d = 4$

(3) $\dfrac{1}{3},\dfrac{5}{3},\dfrac{9}{3},\dfrac{{13}}{3}....$
Here, first term, $a = \dfrac{1}{3}$
Common difference, $d = $\[second{\text{ }}term - first{\text{ }}term\]
Or $d = {a_2} - {a_1}$
$ \Rightarrow d = \dfrac{5}{3} - \dfrac{1}{3}$
$ \Rightarrow d = \dfrac{4}{3}$

(4) $0.6,1.7,2.8,3.9,....$
Here, first term, $a = 0.6$
Common difference, $d = $\[second{\text{ }}term - first{\text{ }}term\]
Or $d = {a_2} - {a_1}$
$ \Rightarrow d = 1.7 - 0.6$
$ \Rightarrow d = 1.1$

Note:In an AP, every succeeding term is obtained by adding $d$ to the preceding term. So, $d$ can be found by subtracting any term from its succeeding term. To obtain $d$ in a given AP, we need not find all of ${a_2} - {a_1},{a_3} - {a_2},{a_4} - {a_3},....$. It is enough to find only one of them .