Understanding the Sequence 14 28 42 56

When dealing with sequences, a crucial step is to identify the pattern followed by the numbers. In the sequence 14 28 42 56, the common difference between consecutive terms is 14. This type of sequence is known as an arithmetic sequence, where each term increases by a fixed number, which in this case is 14.

Identification of the Pattern

The common difference approach can help us find the next number in the sequence:

Common Difference: 28 - 14 14 Common Difference: 42 - 28 14 Common Difference: 56 - 42 14

By consistently adding 14 to the previous number, we can determine that the next term in the sequence would be 56 14 70.

General Formula

The pattern can also be described using a formula for the nth term in the sequence. Given the first term a1 14 and the common difference d 14, the nth term of the sequence can be represented as:

an a1 (n-1)d

For the next term (n 5), the formula becomes:

a5 14 (5-1) * 14 56 56 70

Extensions of the Sequence

The sequence can be extended further by continuing to add the common difference of 14:

70 70 14 84 84 14 98 98 14 112 112 14 126

Thus, the sequence can be written as: 14, 28, 42, 56, 70, 84, 98, 112, 126, ….

Multiplication-Based Pattern

Another way to view this sequence is through multiplication. Each term in the sequence can be expressed as a multiple of 14:

14 × 1 14 14 × 2 28 14 × 3 42 14 × 4 56 14 × 5 70

Errors and Verification

When solving such problems, it's crucial to verify your answer. Let's check the last term of the sequence (56) to see if it follows the pattern:

56 - 42 14 42 - 28 14 28 - 14 14

To find the next term, add the common difference:

56 14 70

Conclusion

In summary, the next number in the sequence 14 28 42 56 is 70. This is derived by consistently adding 14 to the previous term or by multiplying the sequence number by 14. Understanding and applying these methods will help in solving similar problems efficiently.