The given series is 76, 80, 88, 95, 100, 101. While it might appear random at first glance, there's a hidden pattern that governs its progression. Let's dissect the series to uncover the underlying logic and determine the next term.
Unveiling the Pattern:
To find the next term in the series, we need to identify the rule that governs the sequence. Analyzing the differences between consecutive terms, we notice a recurring pattern:
- 76 to 80: An increase of 4
- 80 to 88: An increase of 8
- 88 to 95: An increase of 7
- 95 to 100: An increase of 5
- 100 to 101: An increase of 1
The differences between consecutive terms are 4, 8, 7, 5, and 1. Now, let's examine these differences:
- 4 to 8: An increase of 4
- 8 to 7: A decrease of 1
- 7 to 5: A decrease of 2
- 5 to 1: A decrease of 4
Observe that the differences between the differences (which we can call "second differences") are decreasing by 1 each time: 4, -1, -2, -4. This indicates a quadratic pattern within the original series.
Deriving the Formula:
We can now express this quadratic pattern with a formula. Let's assume the general term of the series is represented by the variable 'T', and the position of the term is denoted by 'n'. We can write the formula as:
T = an² + bn + c
Where 'a', 'b', and 'c' are constants.
To find these constants, we can substitute values from the given series into the formula:
- For n = 1, T = 76: 76 = a(1)² + b(1) + c
- For n = 2, T = 80: 80 = a(2)² + b(2) + c
- For n = 3, T = 88: 88 = a(3)² + b(3) + c
Solving this system of equations, we get:
- a = -1
- b = 6
- c = 71
Therefore, the formula for the series is:
T = -n² + 6n + 71
Determining the Next Term:
Now that we have the formula, we can find the next term by substituting n = 6:
T = -(6)² + 6(6) + 71 = -36 + 36 + 71 = 71
Therefore, the next term in the series is 71.
Conclusion:
By meticulously analyzing the differences between terms, we discovered a quadratic pattern in the series. This pattern allowed us to derive a formula that accurately represents the series and enabled us to determine the next term. The next term of the series 76, 80, 88, 95, 100, 101 is 71.
