Sequences — term to term rule and nth term

A sequence is defined as an ordered list of items — numbers, shapes, letters, or objects — that follow a predictable rule. The key features are that it is ordered and rule-based.

The two essential features are: (1) the list is ordered (position matters), and (2) the items follow a predictable rule (nothing is random).

The option 'An ordered list of items that follow a predictable rule' captures both essential features perfectly.

We need the best definition of a mathematical sequence. Look for the option that captures what makes a sequence different from just any list of numbers.

'A random collection of numbers with no particular order' is the opposite of a sequence — sequences are specifically not random and not unordered.

'Any set of numbers arranged from smallest to largest' describes sorting, not a sequence. A sequence like 10, 7, 4, 1 decreases, and that is still a valid sequence.

'A list of numbers that always increases by the same amount' is too restrictive — sequences can decrease, multiply, or use other rules. This only describes one specific type.

'An ordered list of items that follow a predictable rule' is the only option that correctly captures both the ordered nature and the rule-based structure of a sequence.

The first term of the sequence is 8. Any term-to-term rule must start with this value.

Calculate the difference between consecutive terms: $13-8=5,18-13=5,23-18=5$ The difference is always +5.

Check: $8+5=13$ ✓, $13+5=18$ ✓, $18+5=23$ ✓. The rule works for all consecutive pairs.

The term-to-term rule is: Start at 8 and add 5 each time.

We need to find the starting value and the repeated operation. The first term is clearly 8, so any rule not starting at 8 can be eliminated.

'Start at 13 and add 5 each time' is wrong because 13 is the second term, not the first. The sequence begins at 8.

'Start at 8 and multiply by 5 each time' would give $8,40,200,...$, which does not match the given sequence at all.

'Start at 8 and add 4 each time' would give $8,12,16,20,...$, but the second term of our sequence is 13, not 12. So adding 4 is incorrect.

'Start at 8 and add 5 each time' correctly gives $8,13,18,23,28,...$, matching the sequence perfectly.