Depth · one concept from first principles · step 2 of 4
Build values up from the sum rule
The variables and are both natural numbers, and the function satisfies the two conditions and . Find the value of .
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Your answer
How to solve
Strategy Identify Subproblems — The rule turns any input into a sum of smaller inputs. Split the job into small pieces: get f(2) from 1+1, then f(3) from 2+1, then f(5) from 3+2, each time substituting known values into the rule.
1STEP 1
Find f(2)
Put a=1 and b=1 into the rule.
f(2)=f(1)+f(1)+2×1×1=3+3+2=8
2=1+1, and both pieces are values we already know.
8.F.A.1Convert To Algebra2STEP 2
Find f(3)
Put a=2 and b=1 into the rule.
f(3)=f(2)+f(1)+2×2×1=8+3+4=15
Each new value is built from ones found earlier.
6.EE.A.2Convert To Algebra3STEP 3
Find f(5)
Put a=3 and b=2 into the rule, so we jump straight to 5.
f(5)=f(3)+f(2)+2×3×2=15+8+12=35
5=3+2 uses two values we already have, so there is no need to find f(4).
6.EE.A.2Identify Subproblemscan be built in one step as .
Why?
The rule holds for every pair of natural numbers, so it stays true when and , and then .
🧱A rule true for all numbers is true for eachIf an equation holds for every number, it holds for whatever particular numbers you choose to put in, so special inputs like 0 or 1 are allowed and often reveal hidden values.
Why?
The right side uses only and , which were already found from by the same rule.
Why?
Each found value is equal to the number it was built from, so putting it in place of or keeps the equation true.
🧱Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.
4STEP 4
Subtract
Now compute the difference.
f(3)-f(5)=15-35=-20
The function grows, so f(5) is bigger than f(3) and the difference is negative.
6.EE.A.2Identify SubproblemsAnswer
-20
Reach f(5) another way: f(4)=f(2)+f(2)+2×2×2=8+8+8=24, and f(5)=f(4)+f(1)+2×4×1=24+3+8=35. Same value.
Takeaway
When a rule builds big inputs out of small ones, climb up from the value you know.
- Find f(2)
- Find f(3)
- Find f(5)
- Subtract
Where next?
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