SEQUENCES AND SERIES

Mathematically, Series and sequence are not exactly same, they differ slightly.

"Sequence" is a string of numbers or objects that follows a particular pattern.

"Series", in simple terms, is nothing but the sum of the terms of a sequence.

Series formation involves formation of series and sequences both.

In the questions of this type, a particular pattern is present in the sequence of numbers, symbols or letters or their combination. You are required to identify the pattern present in the sequence and then apply the same to find the missing term.

Sequence can be broadly categorized into three types:

A. Number Sequence

B. Letter Sequence

C. Symbol Sequence

A. Number Sequence

These are again divided into:

1. Arithmetic Sequence

Successive numbers are obtained by adding or subtracting a fixed particular number to or from the preceding number.

This fixed number is called the common difference.

Examples:

3, 5, 7, 9, 11 ...

2, 6, 10, 14, 18 ...

28, 25, 22, 19, 16 ...

Example 1: Identify the next term in the sequence 5, 8, 11, 14 ...

(1)15 (2) 16 (3) 17 (4) 18

The arithmetic sequence has a common difference of 3.

5+3=8, 8+3=11, 11+3=14

So, 14+3 gives 17.

2. Geometric Sequence

Successive number is identified by multiplying or dividing a fixed number (can be fraction also) by the preceding number.

This fixed number is called the common ratio of the geometric sequence.

Examples:

3, 9, 27, 51,153 ...

2, 4, 8, 16, 32 ...

200, 20, 2, 0.2 ...

7, -14, 28, -56, 128 ...

Example 2: Identify the next term in the sequence 4050, 2700, 1800, 1200 ...

(1)1800 (2) 800 (3) 1000 (4) 1050

Find the common ratio among all the terms by dividing them.

4050/2700 = 3/2 , 2700/1800 = 3/2 , 1800/1200 = 3/2

Let the next term be x;

Therefore, 1200/x = 3/2 ,

Now cross multiplying;

X = (1200 x 2)/3 = 800

3. Arithmetico-Geometric Sequence

Each successive term is formed by first adding a fixed number to the preceding term and then multiplying (or dividing) it by some other fixed number.

The difference of successive terms will be in Geometric Progression.

Examples:

2, 6, 14, 30, 62 ...

2, 10, 26, 58, 163 ...

3, 12, 39, 120, 363 ...

Example 3: Identify next term in the sequence 4, 18, 60, 186 ...

(1) 574 (2) 544 (3) 564 (4)578

A fixed number 2 is added to the previous number. Then the result is multiplied by a fixed number 3

(4 + 2) = 8 x 3 = 18

(18 + 2) = 20 x 3 = 60

(60 + 2) = 62 x 3 = 186

Therefore, next term = (186 + 2) x 3 = 564

4. Geometrico-Arithmetic Sequence

Each successive term is formed by first multiplying (or dividing) a fixed number to the preceding term and then adding it by some other fixed number.

Examples:

2, 8, 26, 80, 242 ...

3, 11, 27, 59, 123 ...

4, 10, 22, 46, 94 ...

Example 4: Identify the next term in the sequence 5, 13, 29, 61 ...

(1) 69 (2) 185 (3) 122 (4) 125

A fixed number 2 is multiplied to the preceding number. Then the result is multiplied by a fixed number 3.

(5 x 2) = 10 + 3 = 13,

(13 x 2) =26 + 3 = 29.

(29 x 2) = 58 + 3 = 61.

Therefore, next term = (61 x 2) + 3 = 122 + 3 = 125.

5. Prime Sequence

The number that can be divided only by one and the number itself are called prime numbers.

The prime sequence is a string of prime numbers starting with any of the prime numbers.

Examples:

5, 7, 11, 13, 17, 19 ...

23, 29, 31, 37, 43 ...

Example 5: Identify the next term in the sequence 7, 11, 13, 17, 19 ...

(1) 20 (2) 21 (3) 22 (4) 23

The sequence shows a set of prime numbers. Therefore, the next prime number will be 23

6. Power Sequence

Every number is the nth power of some successive numbers.

This number can be square, cube or any higher powers of a particular string of numbers.

Examples:

1, 2, 3, 4, 5 ... 1st power sequence

1, 4, 9, 16, 25 ... Square sequence(2nd)

1, 8, 27, 64, 125, ... Cube sequence (3rd)

1, 16, 81, 256 ... 4th power sequence.

Example 6: Identify the next term in the sequence 4, 9, 16, 25 ...

(1) 36 (2) 40 (3) 28 (4) 30

It is a 2nd power sequence. So, the next term would be the square of 6 that is 36

7) Reversal Sequence

Each term after reversed gives us a sequence like any of the above-mentioned sequence.

Examples:

31, 51, 71, 91, 12 ...

40, 90, 61, 52, 63 ...

Example 7: Example 7: Identify the next term in the sequence 72, 46, 521 ...

(1) 523 (2) 527 (3) 612 (4) 623

First of all, reverse each term,

The sequence becomes 27, 64, 125 ...

27 is the cube of 3, 64 is the cube of 4, 125 is the cube of 5.

Certainly, the next number will be the cube of 6 that comes out to be 216. By reversing, it becomes 612.

8. Two-tier Sequence

The difference between the successive terms forms any of the above-mentioned sequence like arithmetic sequence, prime sequence, power sequence etc.

Example 8: Identify the next term in the sequence 1, 3, 7, 13, 24 ...

(1) 28 (2) 29 (3) 34 (4) 33

The sequence is two tier arithmetic sequence. The difference between the consecutive terms is:

3 - 1 = 2, 7 - 3 = 4,

13 - 7 = 6, 24 - 13 = 8

These differences form an arithmetic sequence of 2, 4, 6, 8 ...

The next term in this arithmetic sequence should be 10.

So, the difference between the last number (24) and the next term should be 10.

Let x be the next term.

Then, x - 24 = 10

x = 34

Example 9: Identify the next term in the sequence 5, 8, 13, 20, 31 ...

(1) 38 (2) 44 (3) 34 (4) 33

The sequence is two tier prime sequence.

The difference between the consecutive terms is:

8 - 5 = 3, 13 - 8 = 5, 20 - 13 = 7, 31 - 20 = 11

These differences form a prime number sequence of 3, 5, 7, 11...

The next term in this prime sequence should be 13. So, the difference between the last number (31) and the next term should be 13.

Let x be the next term.

Then, x - 31 = 13

x = 44

9. Twin sequence

This sequence has two sequences combined together to form one sequence.

This sequence can be of any type among those mentioned above.

Examples:

1, 3, 2, 5, 3, 7, 4, 9 ...

2, 5, 4, 15, 8, 45, 16 ...

Example 10: Identify the next number in the sequence 1, 4, 3, 7, 5, 10, 7, 13, 9 ...

(1) 10 (2) 11 (3) 16 (4) 21

The terms in the odd positions and even positions both form arithmetic sequence.

The sequence of odd positions has a common difference of 2 and the sequence of even positions has a common difference of 3.

The sequence of even position with a common difference of 3 is 4, 7, 10, 13...

So, the next term in the sequence becomes 13+3 which is 16.

10. Fibonacci series

The first two terms are 1, 1 and each succeeding term is the sum of the previous two numbers.

That is, 1, 1, 2, 3, 5, 8, 13, 21 ...

2+1=3, 3+2=5, 5+3=8, 8+5=13, 13+8=21

11. Telescoping Sequence

Telescoping sequence does not have a set form.

A telescoping sequence is any sequence in which nearly every term cancels with a preceding or following term.

The best example for a telescoping sequence is:

1/(1 x 2) , 1/(2 x 3) , 1/(3 x 4) , ...

Here the terms in this sequence can be written as follows.

1/(1 x 2) = 1/1 - 1/2 , 1/(2 x 3) = 1/2 - 1/3 1/(3 x 4) = 1/3 - 1/4 ...

Thus, the series comprising of the terms of the sequence can be written as:

1/(1 x 2) + 1/(2 x 3) + 1/(3 x 4) + ... + = (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ...

= 1/1 + (- 1/2 +1/2) + (- 1/3 + 1/3) + (- 1/4 +1/4) + ...

= 1 + 0 + 0 + 0 + ... = 1

B. Letter Sequence

In this kind of sequence, a number of letters are arranged in a sequence and we are supposed to find the next letter or letters in the sequence.

For this, convert the letters into numbers.

That is, A = 1, B = 2, C = 3 ... X = 24, Y = 25, Z = 26.

After this conversion, the sequence becomes same like number sequence.

Note: If there occurs a number, say x, greater than 26, subtract 26 from x to get the required number.

In addition, if there is a negative number, add 26 to it to get the required number.

Remember (J = 10, O = 15 and T = 20) for convenience

Example 11: In the adjacent figure, find the missing letter in the box.

(1) M (2) N (3) O (4) P

Convert the letters into number equivalents. The number 15 corresponds to letter O.