FIND THE MISSING TERMS IN THE SEQUENCE

To find the missing terms in the sequence, we have to understand the pattern first.

Types of sequence,

i)  Arithmetic sequence

ii)  Geometric sequence

iii)  Special sequence

Arithmetic sequence :

If common difference is same, then the sequence is known as arithmetic progression. To find nth term of the arithmetic progression, we use the formula

an = a + (n - 1)d

Geometric sequence :

If common ratio is same, then the sequence is known as geometric progression. To find nth term of the geometric progression, we use the formula

an = arn - 1

Problem 1 :

Find the next three terms :

3, 10, 17, 24, 31, ____, ____, ____

Solution :

a = 3

d = 10 - 3 = 7

an = a + (n - 1)d

a6 = 3+(6-1)7

= 3+35

a6 = 38

a7 = 3+(7-1)7

= 3+42

a7 = 45

a8 = 3+(8-1)7

= 3+49

a8 = 52

So, next three terms are 38, 45 and 52.

Problem 2 :

Find the 25th term :

53, 50, 47, 44, 41,…_____

Solution :

a = 53

d = 50 - 53 = -3

an = a + (n-1)d

a25 = 53 + (25-1)(-3)

= 53 - 72

a25 = -19

Hence, 25th term is -19.

Problem 3 :

Find the 20th term :

25, 40, 55, 70, 85,…_____

Solution :

a = 25

d = 40 - 25 = 15

an = a + (n - 1)d

a20 = 25 + (20 - 1)(15)

= 25 + 285

a20 = 310

Hence, 20th term is 310.

Problem 4 :

Find the 75th term 

88, 81, 74, 67, 60,…_____

Solution :

d = 81 - 88 = -7

an = a + (n - 1)d

a75 = 88 + (75 - 1)(-7)

= 88 - 518

a75 = -430

Hence, 75th term is -430.

Determine if the sequence is arithmetic or geometric, and then find the given term.

Problem 5 :

Find the 11th term:

5, 3, 1, -1,… 

Solution :

a = 5

d = 3 - 5 = -2

an = a + (n - 1)d

a11 = 5 + (11 - 1)(-2)

= 5 - 20

a11 = -15

Hence, 11th term is -15.

Problem 6 :

Find 23rd term of 

0.1, 0.15, 0.2, 0.25,… ______

Solution :

a = 0.1

d = 0.15 - 0.1 = 0.05

an = a + (n - 1)d

a23 = 0.1 + (23 - 1)(0.05)

= 0.1 + 1.1

a23 = 1.2

Hence, 23rd term is 1.2.

Problem 7 :

Find 6th term:

25, 75, 225, 675,… ______

Solution :

a = 25

r = 75/25

r = 3

an = a ∙ rn-1

a6 = 25(3)6-1

= 25(3)5

= 25 × 243

a6 = 6075

Hence, 6th term is 6075.

Problem 8 :

Find 22nd term: -2, -5, -8, -11, -14,… ______

Solution :

a = -2

d = -5 - (-2) = -3

an = a + (n - 1)d

a22 = -2 + (22 - 1)(-3)

= -2 - 63

a22 = -65

Hence, 22nd term is -65.

Problem 9 :

Find 10th term: a1 = 320, r = 0.5

Solution :     

a1 = 320, r = 0.5 = 1/2

an = a1 ∙ rn-1

a10 = 320 ∙ (1/2)10 -1

= 320(1/2)9

= 320/512

a10 = 0.625

Hence, 10th term is 0.625.

Problem 10 :

Find 50th term: -9, 2, 13, 24, 35,…

Solution : 

a = -9

d = 2 - (-9) = 11

an = a + (n - 1)d

a50 = -9 + (50 - 1)(11)

= -9 + 539

a50 = 530

Hence, 50th term is 530.

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