91Ó°ÊÓ

A sequence pattern is the recurring feature in a sequence that defines how each term is generated. In the given exercise, the pattern is identified by observing the consistent change from one term to the next. For our sequence: $$2, 7, 12, 17, 22, \text{...}$$, there is a clear and constant increase of 5. This is determined by subtracting any pair of consecutive terms:

• \begin{align*} 7 - 2 = 5 \ 12 - 7 = 5 \ 17 - 12 = 5 \ 22 - 17 = 5 \ \text{...} \ \text{(pattern continues similarly)} \ \text{...} \ \text{(next term)} = 27 \text{ (after adding 5 to 22)} \ \text{(continue the same way)} \ \text{...} \ 32 = 27 + 5 \ 37 = 32 + 5 \ \text{and so on...} \ \text{They all yield an increase of } 5 \text{ consistently.} \ \text{This makes it a straightforward arithmetic sequence.} \ \text{Using this constant difference, you can easily predict future terms in the sequence.} \ \text{You just need to add 5 repeatedly.} \ \text{So the sequence pattern is the key to decoding and extending sequences.} \ \text{Essentially, always trace the common change.} \end{align*}

Consecutive terms refer to the successive elements appearing in a sequence one right after another. In other words, any two terms that follow each other directly forms a pair of consecutive terms. For instance, in the sequence from our exercise: $$2, 7, 12, 17, 22, \text{...}$$,

• The pairs of consecutive terms are: \begin{align*} 2 \text{ and } 7 \ 7 \text{ and } 12 \ 12 \text{ and } 17 \ 17 \text{ and } 22 \ 22 \text{ and } 27 \ \text{...etc.} \ \text{In each case, moving from one term to its consecutive one we simply add 5 (the common difference in our sequence).} \ \text{The understanding of consecutive terms is fundamental to properly identify the pattern within a sequence.} \ \text{By looking at two consecutive terms and the difference, it is possible to derive the rule governing the entire sequence.} \ \text{Noting how consecutive terms relate helps in quick prediction of future values.} \ \text{Always comparing terms next to each other clarifies the steps in pattern formation.} \end{align*}

Addition in the context of arithmetic sequences refers to the process used to find the next term by adding a fixed number to the previous term. This fixed number is known as the common difference in the sequence. Let's review the sequence from the given exercise again: $$2, 7, 12, 17, 22, \text{...}$$. The process works like this:

• Starting from 2, we add 5: \begin{align*} 2 + 5 = 7 \ \text{Next, add 5 to 7:} \ 7 + 5 = 12 \ \text{Continue adding 5 to 12:} \ 12 + 5 = 17 \ \text{... and so on.} \ \text{For the given sequence, this consistent addition results in:} \ 17 + 5 = 22 \ 22 + 5 = 27 \ 27 + 5 = 32 \ 32 + 5 = 37 \ \text{The new terms are always produced by simply continuing the same fixed addition.} \end{align*}

Addition here makes the underlying pattern of the sequence easy to extend and predict accurately. This simple rule - adding the common difference - is a fundamental aspect of arithmetic sequences allowing for straightforward calculations and understanding. Understanding this addition principle simplifies working with larger or more complex sequences as well, ensuring clarity and accuracy in identifying sequence progressions.