Contents

**Basic Concepts of Number Series with Examples**

Number Series is an arrangement of numbers according to certain rule or pattern. In a number series, each number is called ‘Term’ of the series. In an Number Series question, you typically need to identify the,

- Next number in a given number series.
- Missing number in a given number series.
- Incorrect number in a given number series.

**Solving Problems Based on Number Series**

**Step-I:** Observe if there are there any familiar numbers in the given series. For example familiar numbers are Primes Numbers, Perfect Squares, Cubes. Such numbers are very easy to identify.

**Step-II:** Calculate the differences between the numbers. Observe the pattern in the differences. If the differences are growing rapidly it might be a Square Series, Cube Series, or Multiplicative Series. If the numbers are growing slowly it is an arithmetic series. If the differences are not having any pattern then

- It might be a double or triple series. Here every alternate number or every 3rd number form a series.
- It might be a sum or average series. Here sum of two consecutive numbers gives 3rd number or average of first two numbers gives next number.

**Step-III:** Sometimes number will be multiplied and will be added with another number, so we need to check such patterns.

**Prime Number Series**

**Example-I:** 2, 3, 5 ,7, 11, 13, ….

**Answer:** The given series is prime number series. The next prime number is 17.

**Example-II:** 2, 5, 11, 17, 23, …., 41.

**Answer:** The alternate prime numbers are written. The primes after 23 are 29, 31, 37 and 41. So, based on the pattern the answer is 31.

**Difference Series**

**Example-I:** 2, 5, 8, 11, 14, 17, …., 23.

**Answer:** The difference between the numbers in ascending order is 3.

∴ the next term is (17+3 = 20).

**Example-II:** 45, 38, 31, 24, 17,…, 3.

**Answer:** The difference between the numbers in descending order is 7.

∴ the next term is (17-7=10)

**Multiplication Series**

**Example-I:** 2, 6, 18, 54, 162, …., 1458.

**Answer:** Each term is multiplied by 3 to get next term.

∴ The next term is (162×3 = 486).

**Example-II:** 3, 12, 48, 192, …., 3072.

**Answer:** Each term is multiplied by 4 to get the next term.

∴ the next term is(192×4 =768).

**Division Series**

**Example-I:** 720, 120, 24, …, 2, 1

**Answer:** 720/6=120, 120/5=24, 24/4=6, 6/3=2, 2/2=1.

∴ The answer is 6.

**Example-II:** 32, 48, 72, 108, …., 243.

**Answer:** Number × 3/2 = Next Number. 32×3/2=48, 48×3/2=72, 72×3/2=108, 108×3/2=162, 162×3/2=243.

∴ The answer is 162.

**‘n’ Square Series**

**Example-I:** 1, 4, 9, 16, 25, …., 49

**Answer:** The series is 1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}

∴ The next number is 6^{2}=36;

**Example-II:** 0, 4, 16, 36, 64, …., 144.

**Answer:** The series is 0^{2},2^{2},4^{2},6^{2},8^{2} etc.

∴ The next number is 10^{2}=100.

**‘n’ Square Variants**

**n**^{2}−1 Series

^{2}−1 Series

**Example:** 0, 3, 8, 15, 24, 35, 48, ….

**Answer:** The series is 1^{2}−1,2^{2}−1,3^{2}−1,4^{2}−1,5^{2}−1,6^{2}−1,7^{2}−1 etc.

∴ The next number is 8^{2}−1=63.

**Another Logic:** Difference between numbers is 3,5,7,9,11,13 etc. The next number is (48+15=63).

Remember logic can be different, however answer should be the same.

**n**^{2}+1 Series

^{2}+1 Series

**Example:** 2, 5, 10, 17, 26, 37, …., 65.

**Answer:** The series is 1^{2}+1, 2^{2}+1, 3^{2}+1, 4^{2}+1, 5^{2}+1, 6^{2}+1 etc. The next number is 7^{2}+1=50.

**n**^{2}+n Series (or) n^{2}−n Series

^{2}+n Series (or) n

^{2}−n Series

**Example:** 2, 6, 12, 20, …., 42.

**Answer:**

**Logic-I:** The series is 1^{2}+1, 2^{2}+2, 3^{2}+3, 4^{2}+4 etc. The next number = 5^{2}+5=30.

**Logic-II:** The series is 1×2 ,2×3, 3×4, 4×5, The next number is 5×6=30.

**Logic-III:** The series is 2^{2}−2,3^{2}−3,4^{2}−4,5^{2}−5, The next number is 6^{2}−6=30

**‘n’ Cube Series**

**Example:** 1, 8, 27, 64,125, 216, ….

**Answer:** The series is 13, 23, 33, 43, 53, 63, etc. ?

The missing number is 73=343.

**‘n’ Cube Variants**

**n**^{3}+1 Series or n^{3}−1 Series

^{3}+1 Series or n

^{3}−1 Series

**Example-I:** 2, 9, 28, 65, 126, 217, 344, ….

**Answer:** The series is 1^{3}+1,2^{3}+1,3^{3}+1, etc.

∴ The missing number is 8^{3}+1=513.

**Example-II:** 0, 7, 26, 63, 124, …., 342.

**Answer:** The series is 1^{3}−1, 2^{3}−1, 3^{3}−1 etc

∴ The missing number is 6^{3}−1=215.

**‘n’ Cube Variants**

**n**^{3}+n Series or n^{3}−n Series

^{3}+n Series or n

^{3}−n Series

Example 1: 2, 10, 30, 68, 130, …., 350.

**Answer:** The series is 1^{3}+1, 2^{3}+2, 3^{3}+3, 4^{3}+4, 5^{3}+5, ? , 7^{3}+7 etc. ?

The missing number is 6^{3}+6=222.

**Example-II:** 0, 6, 24, 60, 120, 210, ….

**Answer:** The series is 1^{3}−1, 2^{3}−2, 3^{3}−3, 4^{3}−4, 5^{3}−5, 6^{3}−6, etc. ?

the missing number is 7^{3}−7=336.

**n**^{3}+n Series or n^{3}−n Series

^{3}+n Series or n

^{3}−n Series

**Example-I:** 2, 12 ,36, 80, 150, ….

**Answer:** The series is 1^{3}+1^{2}, 2^{3}+2^{2}, 3^{3}+3^{2}, 4^{3}+4^{2}, 5^{3}+5^{2}, etc.

The missing number is 6^{3}+6^{2}=252

**Example-II:** 0, 4, 18, 48, 100, ….

**Answer:** The series is 13−1^{2}, 2^{3}−2^{2}, 3^{3}−3^{2}, 4^{3}−4^{2}, 5^{3}−5^{2}, etc.

The missing number is 6^{3}−6^{2}=180

**ab, a + b Series**

**Example:** 48,12,76,13,54,9,32,…, where a and b are digits of a number

**Answer:** 4+8=12, 7+6=13, 5+4=9, 3+2=5.

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