Depth · one concept from first principles · step 2 of 4
Regroup a long sum of differences
Compute the following.
Number system
Your answer
How to solve
Strategy Organize Information in More Ways — Adding brackets one by one is slow. Instead reorganize: remove the brackets and collect all the even numbers being added and all the odd numbers being subtracted. Each odd number is just one more than an even number on the other list, and that pattern makes almost everything cancel.
1STEP 1
Count the brackets
The first numbers are the even numbers 2, 4, …, 110.
110 ÷ 2 = 55
There are 55 brackets, so 55 even numbers and 55 odd numbers.
7.NS.A.1Look For A Pattern2STEP 2
Regroup the terms
Addition can be done in any order, so gather the added numbers and the subtracted numbers.
(2+4+…+110) - (3+5+…+111)
Two long lists, each with 55 numbers.
7.NS.A.1Organize Information In More Ways3STEP 3
Match each odd number with an even one
Write each odd number as the even number just below it, plus 1: 3=2+1, 5=4+1, and so on up to 111=110+1.
3+5+…+111 = (2+4+…+110) + 55
The odd list is the even list with an extra 1 added 55 times.
7.NS.A.1Look For A PatternThe subtracted odd numbers add up to the added even numbers plus .
Why?
Each odd number is one more than exactly one even number , so the two lists pair off with numbers each.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The sum can be rearranged into all the even numbers followed by all the s.
Why?
Swapping the order of numbers in a sum does not change it.
🧱Commutativity of additionPutting a with b or b with a piles up the same total — order does not change a sum.
Why?
Regrouping which numbers are added first does not change the sum either.
🧱Associativity of additionAdding three piles, it does not matter which two you push together first.
Why?
Fifty-five s added together make .
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
4STEP 4
Let the lists cancel
Subtracting the odd list takes away the whole even list and 55 more.
(2+4+…+110) - (2+4+…+110) - 55 = -55
Everything cancels except 55 copies of -1.
7.NS.A.1Organize Information In More WaysAnswer
-55
Each bracket's value drops by 4: 107, 103, 99, …, ending at 2-111=-109. The values are evenly spaced, so the average is (107+(-109))/2 = -1, and 55 brackets give 55 × (-1) = -55.
Takeaway
When a sum is long, reorder it so matching pieces cancel.
- Count the brackets
- Regroup the terms
- Match each odd number with an even one
- Let the lists cancel
Where next?
Another one like thissuggested
Step back and solidifysuggested
A parent dashboard for the family lives at sensimlab.com.
The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Lattice / Grid Pattern from Small Cases, Pair-Sum Invariant, Small-Domain Constrained Search. Why the shapes repeat →