Problem
Find the rule linking position to term
A list of numbers grows by the same fixed amount each time: 1, 4, 7, 10, 13, and so on. I need to find what number sits in the 30th position.
Operations
Your answer
How to solve
Strategy Look for a Pattern — The list is a number pattern, so I look for how the position links to the number. By checking the first few terms (an easier related problem) I notice each number is its position multiplied by 3, then made 2 smaller. That rule lets me jump straight to the 30th number without writing out all 30 of them.
1STEP 1
See how the numbers grow
Neighbouring numbers differ by 3 every time, so the rule adds 3 at each step.
4-1=3, 7-4=3, 10-7=3
Spotting that the same amount is added each time is exactly what 4th graders practice when they make and describe number patterns.
4.OA.C.5Look For A Pattern2STEP 2
Link the position to the number using small cases
Position 1 has jumped zero times and position 2 once, so any position has jumped (position - 1) times.
number=1+(position-1) × 3
Trying a few easy positions first turns a scary 'find the 30th' into a clear matching rule between position and number.
5.OA.B.3Solve An Easier Related ProblemEach number equals the starting 1 plus 3 taken (position - 1) times.
Why?
The list starts at 1 and climbs by the same 3 for every step past the first, so reaching a position takes one fewer jumps than the position number.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The number sitting at a position is the starting 1 together with every jump made to get there, and nothing else.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Write the rule as one clean expression
Tidying 1 + (position - 1) x 3 gives position x 3 - 2, and position 5 checks out at 13.
number=position × 3-2
Recording the pattern as a short calculation rule is a 5th-grade skill, and testing it on a known term confirms it is right.
5.OA.A.2Look For A Pattern4STEP 4
Use the rule for position 30
Put 30 in place of the position: 30 x 3 - 2 = 90 - 2 = 88. So the 30th number is 88.
30 × 3-2=90-2=88
Once the rule is found, finding any far-away term is just one multiplication and one subtraction.
4.OA.C.5Look For A PatternAnswer
88
The answer should be a whole number bigger than 13 and follow the 'add 3' pattern. From position 5 (which is 13) to position 30 is 25 more steps, and 25 x 3 = 75, so 13 + 75 = 88. This matches, and 88 is a plausible size for the 30th term, so the answer is reasonable.
Takeaway
Find the rule that ties the position to the number (here: position times 3, minus 2) and you can leap straight to the 30th term without listing them all.
- See how the numbers grow
- Link the position to the number using small cases
- Write the rule as one clean expression
- Use the rule for position 30
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested