The explicit form of the pattern 4, 9, 14, 19 can be derived by examining the differences between consecutive terms. The sequence starts at 4 and each subsequent number increases by 5.
To express this mathematically, we first note down the terms:
- 1st term: 4
- 2nd term: 9
- 3rd term: 14
- 4th term: 19
The difference between the terms is constant:
- 9 – 4 = 5
- 14 – 9 = 5
- 19 – 14 = 5
Since the common difference is 5, this is an arithmetic sequence where the initial term (a) is 4, and the common difference (d) is 5.
The explicit formula for the nth term of an arithmetic sequence can be expressed as:
T(n) = a + (n – 1) * d
Plugging in the known values:
T(n) = 4 + (n – 1) * 5
We can simplify this to:
T(n) = 5n – 1
Therefore, the explicit form of the pattern 4, 9, 14, 19 is T(n) = 5n – 1.