What is the explicit formula for the pattern 4, 9, 14, 19?

The sequence given is 4, 9, 14, 19. To find the explicit formula for this pattern, let’s first observe the differences between consecutive terms:

• 9 – 4 = 5
• 14 – 9 = 5
• 19 – 14 = 5

The difference between consecutive terms is constant (5), indicating that this is an arithmetic sequence.

In an arithmetic sequence, the explicit formula can be expressed as:

a_n = a₁ + (n – 1) * d

Where:

• a_n is the n^th term of the sequence,
• a₁ is the first term of the sequence,
• n is the term number, and
• d is the common difference between terms.

For our sequence:

• a₁ = 4
• d = 5

Plugging these values into the formula gives:

a_n = 4 + (n – 1) * 5

This formula can be simplified to:

a_n = 5n – 1

So, the explicit form of the pattern 4, 9, 14, 19 is:

a_n = 5n – 1

With this formula, you can calculate any term in the sequence by substituting the corresponding value of n.