Series Completion: Finding the Missing Term
The task is to identify the next number in the sequence: 3, 6, 11, 18, 27, ?. This involves finding the underlying mathematical pattern.
Analyzing the Series Pattern
We can analyze the pattern by looking at the differences between consecutive terms:
| Term Position (n) | Series Value | Difference (Value[n+1] - Value[n]) |
| 1 | 3 | - |
| 2 | 6 | \(6 - 3 = 3\) |
| 3 | 11 | \(11 - 6 = 5\) |
| 4 | 18 | \(18 - 11 = 7\) |
| 5 | 27 | \(27 - 18 = 9\) |
| 6 | ? | (To be determined) |
Identifying the Pattern in Differences
Observe the sequence of differences: 3, 5, 7, 9. The difference between each consecutive pair of differences is constant:
- \(5 - 3 = 2\)
- \(7 - 5 = 2\)
- \(9 - 7 = 2\)
The differences are increasing by 2 each time. This indicates an arithmetic progression in the differences.
Calculating the Next Term
To find the next term in the original series, we first predict the next difference. Based on the pattern, the next difference will be:
Next difference = Last difference + 2 = \(9 + 2 = 11\).
Now, we add this calculated difference to the last term of the series (27) to find the missing number:
Missing term = Last term + Next difference = \(27 + 11 = 38\).
Verifying with a Formulaic Approach
An alternative method is to find a direct formula for the \(n\)-th term (\(T_n\)). Let's test if the terms follow the pattern \(T_n = n^2 + 2\), where \(n\) is the position of the term:
- \(T_1 = 1^2 + 2 = 1 + 2 = 3\)
- \(T_2 = 2^2 + 2 = 4 + 2 = 6\)
- \(T_3 = 3^2 + 2 = 9 + 2 = 11\)
- \(T_4 = 4^2 + 2 = 16 + 2 = 18\)
- \(T_5 = 5^2 + 2 = 25 + 2 = 27\)
The formula \(T_n = n^2 + 2\) accurately represents the given series.
To find the 6th term (\(n=6\)):
\(T_6 = 6^2 + 2 = 36 + 2 = 38\).
Conclusion
Both the method of differences and the formula verification show that the missing term in the series is 38. This corresponds to the second option provided.
Similar Questions
Which of the following numbers will replace the question mark (?) in the given series?
3, 7, 5, 61, 363 ?
Which of the following numbers will replace the question mark (?) in the given series?
7, 15, 30, 62, 125, 253, ?
Select the correct option that will fill in the blank and complete the series.
1, 3, 3, 6, 5, 12, 7, 24, 9, 48, 11, ____Which of the following numbers will replace the question mark (?) in the given series?
52, 69, 86, 103, ?, 137
Select the number from among the given options that can replace the question mark (?) in the following series.
73, 70, 64, 55, ?
Which of the following numbers will replace the question mark (?) in the given series?
420, 342, 342, ?, 272
Select the number from among the given options that can replace the question mark (?) in the following series.
64, 72, 81, 91, ?
Select the number from among the given options that can replace the question mark (?) in the following series.
47, 47, 48, 50, ?
Select the number from among the given options that can replace the question mark (?) in the following series.
3360, 4080, ? , 5814
Which of the following numbers will replace the question mark (?) in the given series?
990, 891, 792, ?, 594
Important Questions from Number Series
What will come in place of question mark (?) in the following number series?
2, 5, 11, 23, 44, 77, ?
What will come in place of question mark (?) in the following number series?
31, 32, 36, ?, 61, 86
What will come in the place of question mark (?) in the following number series?
3, 6, 18, ?, 630, 6930
What should come in place of the question mark ‘?’ in the following number series?
60, 40, 50, ?, 180, 460
A series is given with one term wrong. Select that wrong term from the given alternatives.
J12, M24, P48, S96, U192
