Answer :

In mathematics, an **arithmetic progression** (**AP**) or arithmetic sequence is a sequence of numbers such that the **difference** between the consecutive terms is constant.

Let three terms a, b and c are in A.P

Therefore, b - a = c - b

Rearranging we get,

And also nth term of an AP is given by

a_{n} = a + (n - 1)d

where, a = first term

n = number of terms

d = common difference

(i) We know: In AP, middle term is average of the other two terms

Hence, middle term = (2 + 26)/2 = 28/2 = 14

Thus, above AP can be written as 2, 14, 26

(ii) We know: In AP, middle term is average of the other two terms

The middle term between 13 and 3 will be;

(13 + 3)/2 = 16/2 = 8

Now, a_{4} – a_{3} = 3 – 8 = - 5

a_{3} – a_{2} = 8 – 13 = - 5

Thus, a_{2} – a_{1} = - 5

Or, 13 – a_{1} = - 5

Or, a_{1} = 13 + 5 = 18

Thus, above AP can be written as 18, 13, 8, 3

(iii)

We have, a = 5 and a_{4} =

We know, nth term of an AP is

a_{n} = a + (n - 1)d

where a and d are first term and common difference respectively.

Now common difference:

a_{4} = a + 3 d

= 5 + 3d

d =

Hence, using d, 2^{nd} term and 3^{rd} term can be calculated as:

a_{2} = a + d

= 5 +

=

a_{3} = a + 2d

= 5 +

= 8

Therefore, the A.P. can be written as:

(iv) Here, a = - 4 and a_{6} = 6

We know, nth term of an AP is

a_{n} = a + (n - 1)d

where a and d are first term and common difference respectively.

Common difference:

a_{6} = a + 5d

6 = -4 + 5d

5d = 6 + 4 = 10

d = 2

The second, third, fourth and fifth terms of this AP are:

a_{2} = a + d = - 4 + 2 = - 2

a_{3} = a + 2d = - 4 + 4 = 0

a_{4} = a + 3d = - 4 + 6 = 2

a_{5} = a + 4d = - 4 + 8 = 4

Thus, the given AP can be written as: - 4, - 2, 0, 2, 4, 6

(v) Given: Second term = 38 and sixth term = -22

So,

a + d = 38.......(1)

a = 38 - d

a + 5d = -22.....(2)

Putting the value of a in equation 2, we get,

38 - d + 5d = -22

38 + 4d = -22

4d = -22 - 38

4d = -60

d = -15

Putting the value of d in equation 1, we get,

a - 15 = 38

a = 38 + 15 = 53

Therefore, the series is

53, 38, 23, 8, -7, -22

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