Problem
Sum an evenly spaced sequence
Start at 4 and keep adding 3. The list runs 4, 7, 10, 13, 16, …. It stops at 100. Add every number in the list.
Reasoning · Grade 4-1 Number Sequences
Your answer
Watch out 96 ÷ 3 = 32 looks like the count, but counting the first number gives 33.
How to solve
Strategy Look for a Pattern — Adding more than thirty numbers one at a time invites a slip, so instead I use the even spacing. I first count the numbers carefully, since everything else depends on that count. Then I try the pairing trick on a tiny four-number version of the same list, where I can check it by plain addition, and only then apply it to the real list. Writing the list forwards and backwards is what makes every pair come out the same, and that is the pattern doing the work.
1STEP 1
Confirm the rule of the list
Every neighbouring gap is 3.
7-4 = 10-7 = 13-10 = 16-13 = 3
Checking several gaps rather than just the first one makes sure the rule really is 'add 3' before anything is built on top of it.
4.OA.C.5Look For A Pattern2STEP 2
Count how many numbers there are
96 ÷ 3 plus one makes 33 numbers.
100 - 4 = 96, 96 ÷ 3 = 32, 32 + 1 = 33
It is the fence-post idea: 32 gaps need 33 posts, so remembering to add one is what keeps the count from being short by exactly one.
3.OA.A.3Make A Systematic List3STEP 3
Test the pairing trick on a short list
Try the pairing trick on a short list first.
4 × 17 = 68, 68 ÷ 2 = 34
A four-number version can be checked by ordinary addition in a few seconds, which is what makes it safe to trust the same trick on thirty-three numbers.
3.OA.D.9Solve An Easier Related Problem4STEP 4
Pair the real list the same way
Pairing the ends, every pair adds to 104.
4+100 = 7+97 = 10+94 = 104
The 3 that one number gains is exactly the 3 the other number loses, so the pair total never budges — that is why the pairing trick works for any evenly spaced list.
3.OA.D.9Look For A PatternPairing the list from both ends gives the same total every time, because what one partner gains the other loses.
Why?
Moving inward raises one partner by the step and lowers the other by the same step, so every pair lands on one fixed total.
🧱Pairing from both ends gives a constant sumIn an evenly spaced list the smallest with the largest, the next with the next, always make the same total, so a long sum becomes a multiplication.
Why?
The rise and the fall are the same size and cancel exactly, which is why the pair total never drifts along the list.
🧱What one part gains the other losesMove some amount from one addend to the other and the total does not budge, so you may make one number round for free.
5STEP 5
Multiply, then halve
Multiply and halve: 104 × 33 ÷ 2 = 1716.
104 × 33 = 3432, 3432 ÷ 2 = 1716
Writing the list twice, once each way, is a fair trade: it doubles the answer but makes every column identical, and undoing the doubling at the end costs only one halving.
4.NBT.B.5Look For A Pattern6STEP 6
Cross-check with the middle number
Checking with the middle number 52 also gives 1716.
4 + 3 × 16 = 52, 52 × 33 = 1716
Half of the pair total 104 is 52, the middle number, so the two methods are the same idea seen from different sides — and getting the same number both ways catches any arithmetic slip.
4.NBT.B.5Look For A PatternAnswer
1716
104 × 33 ÷ 2 = 1716
The 33 numbers run from 4 up to 100 and average about 52, so the total should be near 33 x 50 = 1650; 1716 is right beside that. It also has to be bigger than 33 x 4 = 132 and smaller than 33 x 100 = 3300, and it is. The answer is a plain sum of numbers with no units attached, and it is a whole number as it must be.
Takeaway
Count the numbers first, then pair the smallest with the largest — in an evenly spaced list every pair has the same total, so the whole sum is one multiplication and one halving.
- Confirm the rule of the list
- Count how many numbers there are
- Test the pairing trick on a short list
- Pair the real list the same way
- Multiply, then halve
- Cross-check with the middle number