In the perfect cube series, each term of the series is the perfect cube of some number. These numbers whose cubes form the series are also arranged in the form of a sequence. In this way, you can get a two-stage type perfect cube series. In the following article, we will see different methods to identify the perfect cube series and try and develop various shortcuts and tricks that will help us not only save time but also develop our accuracy. Let us begin the study of the perfect cube series.

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## Perfect Cube Series

First of all, let us begin by writing the cubes of the first 20 numbers. These you will have to commit to the memory. After this, we will try to construct a few perfect cube series and see what else can we do with the concept. The cubes of the first twenty numbers are:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |

11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

1331 | 1728 | 2197 | 2744 | 3375 | 4096 | 4913 | 5832 | 6859 | 8000 |

In the number series normally the series numbers till 30 will occur. Once you detect the fact that the numbers form a cube series, you can easily predict the missing number or find the wrong number in the series. The numbers will not be in the natural sequence but will be mixed around. The first step is to detect that all the members of the series form a cube of some number. The second step is to determine the difference between the cubes or the numbers of which the terms form a cube for. Let us understand this with the help of a few examples.

### Solved Examples WithÂ Cube Series

Example 1: In the following series, a number is such that it does not belong to the arrangement. Select the option that has this number:

8000, 27000, 64000, 105000

A) 8000Â Â Â Â B) 27000Â Â Â Â Â C) 64000Â Â Â Â Â D) 105000

Answer: Focus on the first digit of the numbers. The first digit in each term is a cube of some number except the last option. 105 isn’t the cube of any number, thus the option to select here is D) 105000. The correct entry should be 125000.

Example 2: In the sequence that is present below, one of the terms is missing. Estimate which term is missing and complete the series by selecting it from the options that are present below:

1, 512, 2197, 6859

A) 1, 64, 512, 2197, 6859Â Â Â Â Â Â B)Â 1, 512, 1331, 2197, 6859Â Â Â Â Â C)Â 1, 512, 2197, 4913, 6859Â Â Â Â Â D)Â 1, 216, 512, 2197, 6859

Answer: Here the series has been given but we are not told where the missing number has to go. This makes it a bit more difficult than the other series. But since this is a cube series, we can still predict the number easily. if you take a look at the table of cubes that we wrote above, you will see that the series can be written as follows:

13, 83, 133, 193. Thus the pattern is clear. The first term is the cube of 1 and then the rest of the terms are cubes of 8, 13, and 19. The difference between 13 and 8 is 5 and that between 19 and 13 is 6. Thus there should be a term between 1 and 8 whose difference from 8 is 4 and from 1 is 3. This term is 4 and the cube of 4 is 64. Thus the correct series is A)Â 1, 64, 512, 2197, 6859.

**Browse more Topics under Number Series**

- Perfect Square Series
- Geometric Series
- Two Stage Type Series
- Mixed Series
- Missing Number Series
- Wrong Number Series
- Order and Ranking
- Decimal Fractions
- Square Roots and Cube RootsÂ
- Simplification on BODMAS Rule
- Chain Rule
- Heights & Distances
- Odd Man Out Series
- Number Series Practice Questions

## Mixed Series and Cubes

Sometimes the concept of pure cube series is mixed with the other concepts about series like the Two-tier series. We will see some examples of such series here.

Example 3: An A.P. has 3 as its first term and d = 3. A pure cube series that can be represented by this A.P. is:

A)Â 27,Â 216,Â 729, 1628

B) 3, 216, 729, 1728

C) 27, 216, 729, 1728

D) 3, 6, 9, 12

Answer: First let us try to generate the A.P. Since only four terms are given in the options, we will only generate four terms to optimise time. The A.P. has its first term as 3 and the common difference or ‘d’ = 3 also. We can write the A.P. as 3, 6, 9, 12. Thus taking the cube of each term will give us a series C)Â 27, 216, 729, 1728, which is the right answer.

Example 4: A two-tier cubic series beginning with 1, that has an A.P. with a = 10 and d = 9 as its base series is which of the following:

A) 1,Â 1331, 27000, 195112

B) 1000, 19000, 28000, 37000

C) 1000, 7869, 20804, 50653

D) 1000,Â 6859, 21952, 50653

Answer: the first step again is to determine the A.P. The A.P. has 10 as the first term and the common difference is = 9. Therefore, the series is 10, 19, 28, 37. Now you might want to find the cube of this series in which case you will see that D) is the answer, but that would be wrong. This is so because the question says that the series we ought to find is a two-tier series that begins with 1. So the series is 1, 1 + 10, 11 + 19, 30 + 28 [ we will only take four terms]. The answer to the questions is thus the cube of this series which is A)Â 1,Â 1331, 27000, 195112.

## Practice Questions

Q 1: In the following series, all the terms have been arranged as per the rules of the number series formation. Pick the one with the incorrect entry:

A)Â 2744,Â 4096,Â 5832,Â 6859

B)Â 1,Â 64,Â 1728, 13824

C)Â 3375, 4096, 4913, 5832

D)Â 1, 8, 27, 64

Ans:Â A)Â 2744,Â 4096,Â 5832,Â 6859

Q 2: In a geometric Progression series, the first term is 3 and r = 4. Which of the following can be obtained from this series:

A) 27, 91, 2354, 6785

B) 27, 5832, 6859, 8746

C) 27, 1728, 110592, 7077888

D) 27, 88765, 988786, 10992379, 9013745

Ans:Â C) 27, 1728, 110592, 7077888

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