Question

For the following APs, write the first term and the common difference:
(i) $3,1, - 1, - 3,....$
(ii) $ - 5, - 1,3,7,....$
(iii) $\dfrac{1}{3},\dfrac{5}{3},\dfrac{9}{3},\dfrac{{13}}{3},....$
(iv) $0.6,1.7,2.8,3.9,....$

Answer 1

Hint: Take part (i) into consideration. Start with writing down the first term of the AP which is the starting term of the given sequence in each part. After that, find the common difference by subtracting the second term from the first term or by subtracting the third term from the second term. Repeat this process separately for each of the parts (ii), (iii) and (iv).

Complete step-by-step answer:
Here in this problem, we are given four different arithmetic progressions and we need to find the first term and the common difference for these progressions.
 Before starting with the solution let’s understand a few concepts of Arithmetic progression (AP). In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
On considering (i) for solving first
Here we have the first term of the AP $ = 3$
The common difference can be found using the definition of AP or by simply subtracting the second term by first or third term by second.
$ \Rightarrow {\text{The common difference of AP}} = {2^{nd}}term - {1^{st}}term = {3^{rd}}term - {2^{nd}}term = 1 - 3 = - 2$
Therefore, for (i) the first term is $3$ and the common difference is $ - 2$
In the case of (ii), we have:
The first term of the AP is given $ = $ $ - 5$
$ \Rightarrow {\text{The common difference of AP}} = {2^{nd}}term - {1^{st}}term = {3^{rd}}term - {2^{nd}}term = - 1 - \left( { - 5} \right) = - 1 + 5 = 4$
Therefore, for (ii) the first term is $ - 5$ and the common difference is $4$
For part (iii), we have here:
The first term of the AP will be $ = \dfrac{1}{3}$
$ \Rightarrow {\text{The common difference of AP}} = {2^{nd}}term - {1^{st}}term = {3^{rd}}term - {2^{nd}}term = \dfrac{5}{3} - \dfrac{1}{3} = \dfrac{4}{3}$
Therefore, for (ii) the first term is $\dfrac{1}{3}$ and the common difference is $\dfrac{4}{3}$
In part (iv) of the question, we get:
The first term of the AP will be $ = 0.6$
$ \Rightarrow {\text{The common difference of AP}} = {2^{nd}}term - {1^{st}}term = {3^{rd}}term - {2^{nd}}term = 1.7 - 0.6 = 1.1$
Therefore, we got the first term as $0.6$ and the common difference is $1.1$.

Note: Be careful with the signs while finding the common difference. An alternative approach can be taken to find the common difference by subtracting the last term by second last term or by subtracting any number from its successor, i.e. ${a_n} - {a_{n - 1}}$ , where ‘n’ is the position of the term in the AP.