Unit 2 · Topic 14 · Number Play

Sum of Odd Numbers

Hook

Pixel asked for the sum of the first 5 odd numbers: 1+3+5+7+9.

Kabir reached for his fingers to add them one at a time.

Anaya said '25' immediately, without adding a single pair.

Watch the Lesson

The Story

"Add the first 5 odd numbers," said Pixel. "1, 3, 5, 7, 9." Kabir started counting on his fingers, adding one at a time. Anaya answered before he'd added the second term. "25."

"How?" asked Kabir, once he'd caught up and confirmed 1+3+5+7+9 really was 25. "It's always a square number," said Anaya. "There are 5 odd numbers here, so the answer is 5 squared, 25."

Pixel tried a bigger one: the first 8 odd numbers, 1+3+5+7+9+11+13+15. "8 terms," said Anaya, "so 8 squared, 64." Kabir checked by adding all eight, and got exactly 64.

"Why does that work?" Kabir asked. Pixel drew it with dots: one dot, then an L-shaped layer of 3 more dots around it making a 2x2 square, then an L-shaped layer of 5 more making a 3x3 square, then 7 more making a 4x4 square. "Each new odd-numbered layer of dots grows the square by exactly one row and one column — that's exactly why the running total is always a perfect square."

"So the rule is," said Anaya, "the sum of the first n odd numbers is always n squared. No pairing needed this time — you just count how many odd numbers there are, and square that count." Kabir tested it on the first 10 odd numbers: predicted 100, checked by adding, and got exactly 100.

1+3+5+7+9 = 25, which is 5 squared — five odd terms.

So What Just Happened?

The sum of the first n odd numbers (1, 3, 5, ..., up to the nth odd number) always equals n squared (n x n). For example, the first 5 odd numbers sum to 5 x 5 = 25.

This can be seen visually: each new odd number added is exactly the right size to grow a square of dots by one more row and one more column, so the running total is always a perfect square.

Unlike the natural-number or even-number sums, no pairing step is needed here — simply count how many odd numbers are being added (that count is n) and square it.

Sum of the first n odd numbers = n squared.

Remember This

The sum of the first n odd numbers always equals n squared.
Count how many odd numbers you're adding — that count is n — then square it.
Each new odd-numbered layer of dots grows a square by exactly one row and one column.
No pairing step is needed for this pattern, unlike natural or even number sums.

First 8 odd numbers sum to 8 x 8 = 64.

Try It Yourself

Add the first 6 odd numbers, then confirm the total equals 6 squared.

Predict the sum of the first 12 odd numbers without adding, then check by adding them all.

Word Bank

Odd number: A whole number not divisible by 2, like 1, 3, 5, 7.
Square number: A number formed by multiplying a whole number by itself.
nth odd number: The odd number in position n when counting 1, 3, 5, 7... in order.

Questions

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