In the following question, select the missing number from the given series. 7, 21, 63, 189, 567, ?
Finding the Missing Number in a Series
The question asks us to identify the pattern in the given number series and find the next number.
The series is: 7, 21, 63, 189, 567, ?
Analyzing the Number Series Pattern
Let's examine the relationship between consecutive terms in the series:
- The second term (21) compared to the first term (7): \(21 \div 7 = 3\)
- The third term (63) compared to the second term (21): \(63 \div 21 = 3\)
- The fourth term (189) compared to the third term (63): \(189 \div 63 = 3\)
- The fifth term (567) compared to the fourth term (189): \(567 \div 189 = 3\)
From this analysis, it is clear that each term in the series is obtained by multiplying the previous term by 3.
This type of series where each term is found by multiplying the previous term by a constant value is called a geometric progression.
Calculating the Missing Number
To find the missing number, we need to apply the same pattern to the last given term (567).
The missing number will be the fifth term multiplied by 3.
Missing Number = \(567 \times 3\)
Let's perform the multiplication:
\(567 \times 3 = 1701\)
So, the next number in the series is 1701.
Verification with Options
We check the calculated missing number against the provided options:
| Option | Value | Matches Calculation? |
|---|---|---|
| 1 | 1701 | Yes |
| 2 | 1800 | No |
| 3 | 1720 | No |
| 4 | 1781 | No |
Our calculated value, 1701, matches Option 1.
Conclusion: The Missing Number
Following the established pattern of multiplying by 3, the missing number in the series 7, 21, 63, 189, 567, ? is 1701.
Revision Table: Number Series Patterns
Understanding common number series patterns is crucial for solving these types of questions.
| Pattern Type | Description | Example |
|---|---|---|
| Arithmetic Progression (AP) | Adding or subtracting a constant value (common difference). | 2, 5, 8, 11, ... (common difference +3) |
| Geometric Progression (GP) | Multiplying or dividing by a constant value (common ratio). | 7, 21, 63, 189, ... (common ratio \(\times 3\)) |
| Difference Series | Pattern found in the difference between consecutive terms. | 1, 2, 4, 7, 11, ... (differences: 1, 2, 3, 4, ...) |
| Mixed Series | Combination of different patterns or alternating patterns. | 3, 6, 4, 8, 6, 12, ... (alternating \(\times 2\), \(-2\)) |
Additional Information: Geometric Progression Explained
A Geometric Progression (GP) is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
In the series 7, 21, 63, 189, 567, ..., the first term (a) is 7, and the common ratio (r) is 3.
The formula for the nth term of a GP is \(a_n = a \times r^{n-1}\).
- 1st term (\(n=1\)): \(a_1 = 7 \times 3^{1-1} = 7 \times 3^0 = 7 \times 1 = 7\)
- 2nd term (\(n=2\)): \(a_2 = 7 \times 3^{2-1} = 7 \times 3^1 = 7 \times 3 = 21\)
- 3rd term (\(n=3\)): \(a_3 = 7 \times 3^{3-1} = 7 \times 3^2 = 7 \times 9 = 63\)
- 4th term (\(n=4\)): \(a_4 = 7 \times 3^{4-1} = 7 \times 3^3 = 7 \times 27 = 189\)
- 5th term (\(n=5\)): \(a_5 = 7 \times 3^{5-1} = 7 \times 3^4 = 7 \times 81 = 567\)
- 6th term (\(n=6\)): \(a_6 = 7 \times 3^{6-1} = 7 \times 3^5 = 7 \times 243 = 1701\)
This formula confirms our pattern and calculation for the 6th term.
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