Unit 2 · Topic 11 · Number Play
Pixel gave Anaya and Kabir sixty seconds to add up every whole number from 1 to 15.
Kabir started adding one by one: 1+2=3, 3+3=6, 6+4=10... he ran out of time at 11.
Anaya wrote one line, did one multiplication, and was done before he finished counting on his fingers.
"Add every whole number from 1 to 15," said Pixel, starting a timer. Kabir began adding one at a time: 1+2=3, then +3=6, then +4=10, then +5=15... by the time the timer beeped, he was only up to 11.
Anaya had already finished. "I paired them up," she said. "1 and 15 make 16. 2 and 14 make 16. 3 and 13 make 16. Every pair from opposite ends makes the SAME total — 16." She counted the pairs: 1-15, 2-14, 3-13, 4-12, 5-11, 6-10, 7-9 — seven pairs, each worth 16 — with 8 left over in the exact middle, unpaired.
"So the total is 7 pairs times 16, plus the leftover 8," she said. "7 x 16 = 112, plus 8 = 120." Pixel nodded. "That pairing trick has a name — it's how the mathematician Gauss is said to have summed 1 to 100 in seconds as a child. And it turns into a simple formula: for numbers 1 to n, the sum is n times (n+1), divided by 2."
Kabir tried the formula on the same problem: n=15, so 15 x 16 = 240, divided by 2 = 120. "Same answer, way faster." "Exactly," said Pixel. "The pairing trick and the formula are the same idea — pair the smallest with the largest, and every pair happens to add up to exactly n+1, and there are n/2 such pairs (with a middle leftover if n is odd)."
"So I never actually need to add fifteen numbers one at a time again," said Kabir. "One multiplication and one division, and I'm done — no matter how big n gets."
The sum of the first n consecutive natural numbers (1, 2, 3, ..., n) can be found instantly using the formula: sum = n x (n+1) / 2. This avoids adding every number one by one.
The formula comes from pairing the smallest number with the largest, the second-smallest with the second-largest, and so on — each such pair adds up to exactly n+1, and there are n/2 pairs in total (with one leftover middle number if n is odd, which the formula already accounts for correctly).
This trick works for ANY starting count n, not just 15 — it scales instantly to sums like 1 to 100 or 1 to 1000 without adding a single extra number by hand.
Use the formula to find the sum of 1 to 20, then check it by pairing 1+20, 2+19, and so on.
Find the sum of the first 50 natural numbers using only the formula.
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