Unit 2 · Topic 12 · Number Play

Sum of Numbers Going Up and Down

Hook

Pixel wrote 1+2+3+2+1 on the board — numbers climbing up, then climbing right back down.

Kabir started adding term by term: 1+2=3, +3=6, +2=8, +1=9.

Anaya just said '9 — that's 3 squared,' before Kabir had even added the third term.

Watch the Lesson

The Story

Pixel wrote a strange-looking sum on the board: 1 + 2 + 3 + 2 + 1. "Notice anything about the pattern?" Kabir squinted. "It climbs up to 3, then comes back down." He started adding left to right: 1+2=3, then +3=6, then +2=8, then +1=9.

Anaya had already called out "9" before he reached the end. "How?" asked Kabir. "It's a square number," she said. "The peak is 3, and the answer is 3 squared, which is 9."

Pixel tried a bigger one: 1+2+3+4+3+2+1. "Peak is 4," said Anaya instantly. "So the answer should be 4 squared, 16." Kabir checked by adding all seven terms one at a time: 1+2+3+4+3+2+1 = 16. It matched.

"Why does that work?" asked Kabir. Pixel drew it as a picture instead: a triangle of dots going up to a row of 4, then a mirrored triangle coming back down, missing the peak row (since it's shared). "Picture two triangular staircases glued together at their tallest step — that shape is exactly a square, side length equal to the peak."

Anaya summarised it: "Going up to n and back down to 1 — like 1+2+...+n+...+2+1 — always totals n squared. You never need to add every term; you just need to know the peak." Kabir tested it once more with a peak of 5: 1+2+3+4+5+4+3+2+1, predicted 25, and checked — 25, exactly right.

1+2+3+2+1 = 9, which is 3 squared — the peak, squared.

So What Just Happened?

A sum that climbs from 1 up to a peak number n and then comes back down to 1 again — written as 1+2+...+n+...+2+1 — always equals n squared (n x n), no matter how large n is.

This works because the up-and-down shape is really two triangular staircases joined at their shared peak, which together fill out exactly a square of side length n.

To find the total instantly, you only need to identify the PEAK value of the sequence and square it — you never need to add every individual term.

1+2+...+n+...+2+1 = n squared.

Remember This

1+2+...+n+...+2+1 (climbing up to n, then back down to 1) always equals n squared.
You only need to find the peak value and square it — no term-by-term addition needed.
The pattern forms two triangular staircases joined at the peak, filling a square shape.
This works for any peak n, no matter how large.

1+2+3+4+5+4+3+2+1 = 25, matching 5 squared.

Try It Yourself

Without adding term by term, predict the total of 1+2+3+4+5+6+5+4+3+2+1, then check by adding.

Write out the up-and-down sum for a peak of 7 and confirm it equals 49.

Word Bank

Peak: The single largest number in an up-and-down sequence.
Square number: A number formed by multiplying a whole number by itself.
Triangular staircase: A shape formed by rows growing by one each time, like 1, 2, 3, 4.

Questions

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