Understanding the Sequence
The sequence you provided is defined by summing specific numbers. Let's break it down:
Sequence Breakdown
- The first term is: 1
- The second term adds 7: 1 + 7 = 8
- The third term adds 19: 1 + 7 + 19 = 27
- The fourth term adds 37: 1 + 7 + 19 + 37 = 64
Identifying the Pattern
- The terms to be added are: 7, 19, 37...
- To identify the pattern, observe the differences between the added numbers:
- 19 - 7 = 12
- 37 - 19 = 18
- The differences (12, 18) indicate that the next difference may increase by 6, suggesting a quadratic relationship.
Finding the Next Terms
- Continuing the pattern:
- Next difference: 37 + (18 + 6) = 61
- Therefore, the next number to be added is 61.
Continuing the Sequence
- The next term would be:
- 1 + 7 + 19 + 37 + 61 = 125
Summary of the Sequence
- The sequence appears to be cumulative sums of the numbers: 1, 7, 19, 37, 61...
- It reflects a quadratic nature due to increasing differences.
This approach provides clarity in understanding how to derive subsequent terms in the sequence, making it easier to tackle similar problems in exams like UPSC.