FINDING THE MISSING TERMS IN AN ARITHMETIC SEQUENCE

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

Formula to find nth term of an Arithmetic Sequence :

an = a + (n - 1)d

a----> nth term

a ----> first term

d ----> common difference

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.

an = a + (n - 1)d

Second Term :

a2a + (2 - 1)d

= a + d

= 5 + 3/2

= (10 + 3)/2

= 13/2

Third Term :

a3a + (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.

an = 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 :

a= a + (5 - 1)d

a= a + 4d

 = -2 + 4(2)

= -2 + 8

= 6

Example 3 :

___ , 38, ___ , ___ , ___, -22

Solution :

a6 = -22

a2 = 38

a + (2 - 1)d = 38

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

a6 = -22

a + (6 - 1)d = -22

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

(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.

an = 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 :

a= a + (5 - 1)d

 = a + 4d

= 53 + 4(-15)

= 53 - 60

= -7

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