For the following arithmetic progressions write the first term a and the common difference d:

Solution:

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 $d=-2$