Understanding the arithmetic sequence 9-14-19 can provide valuable insights into the nature of such sequences. In this article, we will delve into the first 20 terms of this sequence, explore the pattern, and calculate the difference between the first and the 20th term. We will also discuss the broader implications of these calculations.

Introduction to Arithmetic Sequences

An arithmetic sequence is a sequence of numbers such that the difference between any two successive members is constant. This constant difference is known as the common difference.

The Sequence 9-14-19

The given sequence is 9-14-19. Let's calculate the first 20 terms of this sequence step-by-step.

Step 1: Identify the First Term and Common Difference

The first term, a1, is 9.

The common difference, d, is 5 because each term is 5 more than the previous term.

Step 2: Calculate the nth Term Formula

The formula for the nth term of an arithmetic sequence is given by:

an a1 (n - 1) d

Substituting the values for the given sequence:

an 9 (n - 1) 5

Step 3: Calculate the First 20 Terms

a1 9 (1 - 1) 5 9 a2 9 (2 - 1) 5 14 a3 9 (3 - 1) 5 19 a4 9 (4 - 1) 5 24 a5 9 (5 - 1) 5 29 a6 9 (6 - 1) 5 34 a7 9 (7 - 1) 5 39 a8 9 (8 - 1) 5 44 a9 9 (9 - 1) 5 49 a10 9 (10 - 1) 5 54 a11 9 (11 - 1) 5 59 a12 9 (12 - 1) 5 64 a13 9 (13 - 1) 5 69 a14 9 (14 - 1) 5 74 a15 9 (15 - 1) 5 79 a16 9 (16 - 1) 5 84 a17 9 (17 - 1) 5 89 a18 9 (18 - 1) 5 94 a19 9 (19 - 1) 5 99 a20 9 (20 - 1) 5 104

The first 20 terms of the sequence are:

9 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104

Differences Between Consecutive Terms

The difference between consecutive terms in this sequence is consistently 5. This can be verified by subtracting each term from the next:

14 - 9 5 19 - 14 5 24 - 19 5 29 - 24 5 34 - 29 5 39 - 34 5 44 - 39 5 49 - 44 5 54 - 49 5 59 - 54 5 64 - 59 5 69 - 64 5 74 - 69 5 79 - 74 5 84 - 79 5 89 - 84 5 94 - 89 5 99 - 94 5 104 - 99 5

Conclusion: Difference Between the First and the 20th Term

Finally, to find the difference between the first and the 20th term, we subtract the first term (9) from the 20th term (104).

104 - 9 95

The difference between the first 20 terms is 95, confirming the consistency of the common difference in the arithmetic sequence.

In conclusion, the arithmetic sequence 9-14-19 exhibits a pattern where each term increases by 5, making the first and the 20th terms 9 and 104, respectively, with a difference of 95.