Select the number from among the given options that can replace the question mark (?) in the following series. 3, 6, 15, 42, 123, ?
366
Understanding the Number Series Pattern
The given number series is 3, 6, 15, 42, 123, ?. We need to find the number that replaces the question mark by identifying the underlying pattern in the sequence.
Analyzing the Differences Between Consecutive Terms
Let's look at the difference between each term and its preceding term:
- Difference between the 2nd and 1st term: $6 - 3 = 3$
- Difference between the 3rd and 2nd term: $15 - 6 = 9$
- Difference between the 4th and 3rd term: $42 - 15 = 27$
- Difference between the 5th and 4th term: $123 - 42 = 81$
The sequence of differences is 3, 9, 27, 81.
Identifying the Pattern in Differences
Let's examine the sequence of differences (3, 9, 27, 81). We can observe that each term is a power of 3:
- $3 = 3^1$
- $9 = 3^2$
- $27 = 3^3$
- $81 = 3^4$
The pattern of differences follows successive powers of 3, starting with $3^1$. The next difference in the series should therefore be the next power of 3, which is $3^5$.
Calculating the Next Difference
The next difference is $3^5$.
Calculation of $3^5$:
$3^5 = 3 \times 3 \times 3 \times 3 \times 3$
$3^5 = (3 \times 3) \times (3 \times 3) \times 3$
$3^5 = 9 \times 9 \times 3$
$3^5 = 81 \times 3$
$3^5 = 243$
So, the next difference is 243.
Finding the Missing Term in the Series
To find the next term in the original series, we add the calculated next difference (243) to the last term given in the series (123).
Missing term = Last term + Next difference
Missing term = $123 + 243$
Missing term = $366$
Step-by-Step Solution Summary
Here is a summary of the steps:
- Observe the given series: 3, 6, 15, 42, 123, ?.
- Calculate the differences between consecutive terms: $6-3=3$, $15-6=9$, $42-15=27$, $123-42=81$.
- Identify the pattern in these differences: 3, 9, 27, 81, which are $3^1, 3^2, 3^3, 3^4$.
- Determine the next term in the difference pattern: $3^5$.
- Calculate the value of $3^5$: $3^5 = 243$.
- Add this value to the last term of the original series: $123 + 243 = 366$.
The number that replaces the question mark is 366.
| Term Number | Term Value | Difference from Previous Term | Difference Pattern |
|---|---|---|---|
| 1 | 3 | - | - |
| 2 | 6 | $6 - 3 = 3$ | $3^1$ |
| 3 | 15 | $15 - 6 = 9$ | $3^2$ |
| 4 | 42 | $42 - 15 = 27$ | $3^3$ |
| 5 | 123 | $123 - 42 = 81$ | $3^4$ |
| 6 | ? | $123 + 3^5 = 123 + 243$ | $3^5$ |
Revision Table: Key Concepts for Number Series
Understanding common patterns is crucial for solving number series questions. Here are a few types:
| Pattern Type | Description | Example |
|---|---|---|
| Arithmetic Series | Constant difference between terms. | 2, 5, 8, 11, ... (Difference = 3) |
| Geometric Series | Constant ratio between terms. | 3, 6, 12, 24, ... (Ratio = 2) |
| Difference Series | Differences between terms follow a pattern (arithmetic, geometric, squares, cubes, etc.). Like the problem solved here. | 1, 2, 4, 7, 11, ... (Differences: 1, 2, 3, 4) |
| Mixed Series | Combination of different patterns. | Often involves alternating patterns or sequences related to operations. |
| Fibonacci or Similar Series | Terms are sum/difference of previous terms. | 1, 1, 2, 3, 5, 8, ... (Each term is sum of previous two) |
Additional Information: Solving Quantitative Aptitude Questions
Number series questions are common in quantitative aptitude tests. To improve your ability to solve these:
- Practice identifying patterns quickly.
- Calculate differences between terms, then differences of differences.
- Look for ratios between terms.
- Consider patterns involving squares, cubes, primes, or factorials.
- Check for alternating patterns or combinations of rules.
- Review basic arithmetic operations, powers, and sequences.
Consistent practice with various types of number series will help you recognize patterns more efficiently during exams.
Important Questions from Number Series
Select the number from among the given options that can replace the question mark (?) in the following series.
37, 52, 74, 104, 143, ?
Select the number that can replace the question mark (?) in the following series.
17, 19, 22, 27, 34, 45, 58,?Select the number from among the given options that can replace the question mark (?) in the following series.
10, 14, 31, 35, 73, 77, ?
Select the number from among the given options that can replace the question mark (?) in the following series.
215, 231, 256, 292, ?
Select the number from among the given options that can replace the question mark (?) in the following series.
6, 6, 8, 24, 28, 140, ?
