# Arithmetic Progression

## Exercise 5.2 Part 2

Question: 3 – In the following APs, find the missing terms in the boxes

(i) 2, (...), 26

Solution: We know that in AP, middle term is average of the other two terms

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

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

(ii) (...), 13, (...), 3

Solution: Here, 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) 5, (...), (...), 9½

Solution: We have, a = 5 and a4 = 9½

Now common difference can be calculated as follows:

a_4=a+(4-1)d

Or, 91/2=5+3d

Or, 3d=(19)/(2)-5=9/2

Or, d=9/2xx1/3=3/2

Using common difference, 2nd and 3rd terms can be calculated as follows:

a_2=a+d

=5+3/2=(13)/(2)

a_3=a+2d

=5+2xx3/2=8

Thus, the AP can be written as; 5, 13/2, 8, 19/2

(iv) – 4, (...), (...), (...), (...), 6

Solution: Here, a = - 4 and a6 = 6

Common difference can be calculated as follows:

a_6 = a + 5d

Or, 6 = -4 + 5d

Or, 5d = 6 + 4 = 10

Or, d = 2

The second, third, fourth and fifth terms of this AP can be calculated as follows:

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) (...), 38, (...), (...), (...), - 22

Solution: Let us take 38 as the first term and – 22 as the 5th term

Using this, common difference can be calculated as follows:

a_5 = a + 4d

Or, - 22 = 38 + 4d

Or, 4d = - 22 – 38 = - 60

Or, d = - 15

If 38 is the second term, then first term can be calculated as follows:

a = a_2 – d = 38 + 15 = 53

Third, fourth and fifth terms can be calculated as follows:

a_3 = a + 2d = 53 + 2(- 15) = 53 – 30 = 23

a_4 = a + 3d = 53 – 45 = 8

a_5 = a + 4d = 53 – 60 = - 7

So, the AP can be written as: 53, 38, 23, 8, - 7, - 22

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