For the following arithmetic progressions write the first term a and the common difference d:
(i) −5, −1, 3, 7, ...
(ii) $\frac{1}{5}, \frac{3}{5}, \frac{5}{5}, \frac{7}{5}$
(iii) 0.3, 0.55, 0.80, 1.05, ...
(iv) −1.1, −3.1, −5.1, −7.1, ...
In the given problem, we need to write the first term (a) and the common difference (d) of the given A.P
(i) −5, −1, 3, 7 …
Here, first term of the given A.P is (a) = −5
Now, we will find the difference between the two terms of the given A.P
$a_{2}-a_{1}=-1-(-5)$
$a_{2}-a_{1}=4$
Similarly,
$a_{3}-a_{2}=3-(-1)$
$a_{3}-a_{2}=4$
$a_{4}-a_{3}=7-3$ Also,
$a_{4}-a_{3}=4$
As $a_{2}-a_{1}=a_{3}-a_{2}=a_{4}-a_{3}=4$
Therefore, the first term of the given A.P is $a=-5$ and the common difference of the given is $d=4$
(ii) $\frac{1}{5}, \frac{3}{5}, \frac{5}{5}, \frac{7}{5}, \ldots \ldots$
Here, first term of the given A.P is $(a)=\frac{1}{5}$
Now, we will find the difference between the two terms of the given A.P
$a_{2}-a_{1}=\frac{3}{5}-\frac{1}{5}$
$a_{2}-a_{1}=\frac{2}{5}$
Similarly,
$a_{3}-a_{2}=\frac{5}{5}-\frac{3}{5}$
$a_{3}-a_{2}=\frac{2}{5}$
Also,
$a_{4}-a_{3}=\frac{7}{5}-\frac{5}{5}$
$a_{4}-a_{3}=\frac{2}{5}$
As $a_{2}-a_{1}=a_{3}-a_{2}=a_{4}-a_{3}=\frac{2}{5}$
Therefore, the first term of the given A.P is $a=\frac{1}{5}$ and the common difference is $d=\frac{2}{5}$
(iii) 0.3, 0.55, 0.80, 1.05, …
Here, first term of the given A.P is (a) = 0.3
Now, we will find the difference between the two terms of the given A.P
$a_{2}-a_{1}=0.55-0.3$
$a_{3}-a_{2}=0.25$
Similarly,
$a_{3}-a_{2}=0.80-0.55$
$a_{3}-a_{2}=0.25$
Also,
$a_{4}-a_{3}=1.05-0.80$
$a_{4}-a_{3}=0.25$
As $a_{2}-a_{1}=a_{3}-a_{2}=a_{4}-a_{3}=0.25$
Therefore, the first term of A.P is $a=0.3$ and the common difference is $d=0.25$
(iv) −1.1, −3.1, −5.1, −7.1...
Here, first term of the given A.P is (a) = −1.1
Now, we will find the difference between the two terms of the given A.P
$a_{2}-a_{1}=-3.1-(-1.1)$
$a_{2}-a_{1}=-2$
Similarly,
$a_{3}-a_{2}=-5.1-(-3.1)$
$a_{3}-a_{2}=-2$
Also,
$a_{4}-a_{3}=-7.1-(-5.1)$
$a_{4}-a_{3}=-2$
As $a_{2}-a_{1}=a_{3}-a_{2}=a_{4}-a_{3}=-2$
Therefore, the first term of A.P is $a=-1.1$ and the common difference is $d=-2$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.