FIND THE NEXT THREE TERMS IN EACH SEQUENCE

Find the next three terms of each sequences of each sequence. Next to the problem, describe the pattern in words.

Problem 1 :

4, -16, 64, -256, 1024,…

A) -1024, 4096, -16384          B) -4096, -16384, -65536

C) - 4096, 16384, -65536       D) -1024, -4096, -16384

Solution :

To find the next three, first we have to find out the pattern followed in sequence.

Pattern :

2^nd term ==>  4(-4)  ==>  -16

3^rd term ==>  -16(-4)  ==>  64

4^th term ==>  64(-4)  ==>  -256

5^th term ==>  -256(-4)  ==>  1024

6^th term = -4(1024) = -4096

7^th term = 4(4096) = 16384

8^th term = -4(16384) = -65536

Hence the next three terms are -4096, 16384, -65536.

So, option (C) is correct.

Problem 2 :

1, 5, 25, 125, 625,…

A) 6250, 31250, 156250         B) 3125, 15625, 78125

C) 2/3125, 2/15625, 2/78125       D) 2/625, 2/3125, 2/15625

Solution :

Pattern followed :

Every term is being multiplied by 5.

6^th term = 5(625) = 3125

7^th term = 5(3125) = 15625

8^th term = 5(15625) = 78125

Hence the next three terms are 3125, 15625, 78125.

So, option B) is correct.

Problem 3 :

4, 16, 36, 64, 100,…

A) 144, 196, 281        B) 144, 196, 256

C) 164, 196, 307        D) 144, 196, 300

Solution :

To find the next three, first we have to find out the pattern followed in sequence.

Pattern :

4 = 2²_,  16 = 4²_,  36 = 6²_, 64 = 8²_, 100 = 10²

So, the next three terms will be 12², 14², 16².

Hence the next three terms will be 144, 196, 256.

So, option B) is correct.

Problem 4 :

4, -12, 36, -108, 324,…

A) -972, -2916, -8748              B) -972, 2916, -8748

C) -4/243, 4/729, -4/2187       D) -4/243, -4/729, -4/2187

Solution :

To find the next three, first we have to find out the pattern followed in sequence.

Pattern :

Multiplying the first term by -3, we get the second term. Multiplying the second term by 3, we get the third term.

6^th term = -3(324) = -972

7^th term = 3(972) = 2916

8^th term = -3(2916) = -8748

Hence the next three terms are -972, 2916, -8748.

So, option B) is correct.

Problem 5 :

1, 9, 25, 49, 81,…

A) 121, 169, 225            B) 121, 193, 225

C) 140, 160, 225            D) 167, 160, 225

Solution :

To find the next three, first we have to find out the pattern followed in sequence.

Pattern :

The square of odd numbers is the pattern.

1 = 1²_,  9 = 3²_, 25 = 5²_, 49 = 7²_, 81 = 9²

6^th term = 11² ==> 121

7^th term = 13² ==> 169

9^th term = 15² = 225

Hence the next three terms are 121, 169, 225.

Problem 6 :

Rectangular tables are placed together along their short edges, as shown in the diagram. Write a rule for the number of people that can be seated around n tables arranged in this manner. Then graph the sequence.

Solution :

Number of tables in the first row = 1, number of people can be seated = 4

Number of tables in the second row = 2, number of people can be seated = 8

Number of tables in the third row = 3, number of people can be seated = 12

By converting into coordinates, we get

(1, 4), (2, 8) and (3, 12), .............

Problem 7 :

The right side of the orchestra section of the Nederlander Theater in New York City has 19 rows, and the last row has 27 seats. The numbers of seats in each row increase by 1 as you move toward the back of the section. How many seats are in this section of the theater ?

Solution :

Total number of rows (n) = 19

Number of seats in the last row = 27

Number of seats increase = 1

The pattern creates an arithmetic progression.

a_n = a + (n - 1) d

27 = a + (19 - 1)(1)

27 = a + 18

a = 27 - 18

a = 9

Sn = (n/2) [a + l|

n = 19, l = 27 and d = 1

= (19/2)[9 + 27]

= (19/2) x 36

= 19 x 18

= 342

There are 342 seats in this section in the theater.

Problem 8 :

Find the next three terms in each arithmetic sequence.

a) b, b + 4, b + 8, ..........

b) 2a - 5, 2a + 2, 2a + 9, ...............

Solution :

a) b, b + 4, b + 8, ..........

Since it is arithmetic sequence, the common difference will be the same.

Common difference = b + 4 - b

= 4

4^th term = b + 8 + 4 ==> b + 12

5^th term = b + 12 + 4 ==> b + 16

6^th term = b + 16 + 4 ==> b + 20

So, the next three terms are 

b + 12, b + 16, b + 20, ...............

b) 2a - 5, 2a + 2, 2a + 9, ...............

Common difference = 2a + 2 - (2a - 5)

= 2a + 2 - 2a + 5

= 2 + 5

= 7

4^th term = 2a + 9 + 7 ==> 2a + 16

5^th term = 2a + 16 + 7 ==> 2a + 23

6^th term = 2a + 23 + 7 ==> 2a + 30

So, the next three terms are 

2a + 16, 2a + 23, 2a + 30, .................