The next number in the sequence 7, 14, 28 is 56.
Unveiling the Pattern in the Sequence
To determine the subsequent number in a series like 7, 14, 28, the first step is to identify the underlying mathematical relationship between its existing terms. By examining the given numbers, a clear pattern emerges:
- From 7 to 14: The number 7 is multiplied by 2 to get 14 (7 × 2 = 14).
- From 14 to 28: The number 14 is multiplied by 2 to get 28 (14 × 2 = 28).
This consistent multiplication by a fixed number indicates that this is a geometric sequence. In a geometric sequence, each term after the first is found by multiplying the previous one by a constant value, known as the common ratio. In this particular sequence, the common ratio is 2.
Calculating the Next Term
Given that the pattern involves multiplying each term by 2 to obtain the next, calculating the missing number is straightforward:
- Identify the last known term in the sequence: 28.
- Apply the common ratio: Multiply 28 by 2.
- 28 × 2 = 56
Thus, the sequence progresses as 7, 14, 28, 56, and continues in the same manner (e.g., 56 × 2 = 112, and so on).
Understanding Geometric Sequences
Geometric sequences are a fundamental concept in mathematics with applications in various fields. They are characterized by a constant common ratio, which dictates the rate at which the terms increase or decrease.
Key aspects of geometric sequences include:
- Constant Multiplication: Every term is derived from the previous one by multiplying by the same non-zero number.
- Exponential Growth or Decay: If the common ratio is greater than 1, the sequence exhibits exponential growth. If it's between 0 and 1, it shows exponential decay.
Let's illustrate the progression of this sequence with a table:
| Term Number | Previous Term | Operation | Current Term Value |
|---|---|---|---|
| 1st | N/A | N/A | 7 |
| 2nd | 7 | × 2 | 14 |
| 3rd | 14 | × 2 | 28 |
| 4th | 28 | × 2 | 56 |
| 5th | 56 | × 2 | 112 |
This visual representation clearly demonstrates the consistent multiplication by 2, leading directly to the identification of 56 as the next number in the sequence. Understanding such patterns is crucial for solving various mathematical problems and interpreting data series.
