FINDING THE MISSING TERMS IN AN ARITHMETIC SEQUENCE

Using the formula for n^th term of an arithmetic sequence, we can find the missing terms.

Find the missing terms in the following arithmetic sequences.

Example 1 :

5, ___ , ___ , 9½

Solution :

a = 5

a₄ = 9½

a + (4 - 1)d = 19/2

a + 3d = 19/2

Substitute a = 5.

5 + 3d = 19/2

Subtract 5 from both sides.

3d = 19/2 - 5

3d = (19 - 10)/2

3d = 9/2

Divide both sides by 3.

d = 3/2

In the given arithmetic sequence, the missing terms are second term and third term.

a_n = a + (n - 1)d

Second Term :

a₂ = a + (2 - 1)d

= a + d

= 5 + 3/2

= (10 + 3)/2

= 13/2

Third Term :

a₃ = a + (3 - 1)d

= a + 2d

= 5 + 2(3/2)

= 5 + 3

= 8

Example 2 :

-4, ___, ___ , ___ , ___ , 6

Solution :

a = -4

a₆ = 6

a + 5d = 6

-4 + 5d = 6

Add 4 to both sides.

5d = 10

Divide both sides by 5.

d = 2

In the given arithmetic sequence, the missing terms are second, third, fourth and fifth terms.

a_n = a + (n - 1)d

Second Term :

a₂ = a + (2 - 1)d

a₂ = a + d

= -4 + 2

= -2

Third Term :

a₃ = a + (3 - 1)d

a₃ = a + 2d 

= -4 + 2(2)

= -4 + 4

= 0

Fourth Term :

a₄ = a + (4 - 1)d

a₄ = a + 3d

= -4+3(2)

  = -4 + 6 

 = 2

Fifth Term :

a₅ = a + (5 - 1)d

a₅ = a + 4d

 = -2 + 4(2)

= -2 + 8

= 6

Example 3 :

___ , 38, ___ , ___ , ___, -22

Solution :

a₆ = -22

(2) - (1) :

4d = -60

Divide both sides by 4.

d = -15

Substitute d = -15 in (1).

a + (-15) = 38

a - 15 = 38

Add 15 to both sides. 

a = 53

In the given arithmetic sequence, the missing terms are first, third, fourth and fifth terms.

a_n = a + (n - 1)d

First Term :

a = 53

Third Term :

a₃ = a + (3 - 1)d

= a + 2d

= 53 + 2(-15)

= 53 - 30

= 23

Fourth Term :

a₄ = a + (4 - 1)d

= a + 3d

= 53 + 3(-15)

= 53 - 45

= 8

Fifth Term :

a₅ = a + (5 - 1)d

 = a + 4d

= 53 + 4(-15)

= 53 - 60

= -7

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