Sometimes, number sequences can be related to each other in surprising ways.
What happens when we start adding up odd numbers? Let's take a look at the sums of consecutive odd numbers starting from 1.
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?
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Master One Concept in One Sprint
This is a real concept sprint from Relations Among Sequences (Class VI Mathematics).
A chapter usually has 3-10 concept sprints. This demo shows one complete concept flow with interactive blocks.
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content
Adding Up Sequences
Hook showing how adding odd numbers results in squares.
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IMAGE
Visualising Square as Diamond Dot pattern
Diamond dot pattern showing the up and down addition.
Click to zoom
Count the number of dots in each row and add them together. This sum results in a square number:
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content
Adding Up and Down
Worked example showing how counting up and down makes squares.
Problem
Find the resulting number sequence when you start adding the counting numbers up, and then add them back down. Does this relate to any sequence we already know?
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Faded Example
Pairs of Triangular Numbers
Faded step-by-step for adding consecutive triangular numbers.
0 of 3 blanks filled
Let's investigate what happens when we add pairs of consecutive triangular numbers. The first few triangular numbers are 1, 3, 6, 10, and 15.
If we take the first pair, , the sum is .
For the next pair, , the sum is .
Continuing this pattern with , we get 16, and gives us 25. Looking at the resulting sums (4, 9, 16, 25...), we can see a clear and beautiful pattern emerging.
This shows that adding consecutive triangular numbers produces the sequence of numbers.
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quiz
Summing Large Sequences
Practice calculating the sum of 100 odd numbers.
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What is the sum of the first 100 odd numbers ()?
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Refraction of Light Through a Glass Slab · Labeled Diagram
Why does light bend more inside a denser medium?
B
Light slows down when it enters a denser medium. That change in speed bends its direction at the boundary — Snell's law tells you the exact angle from the refractive indices. The denser the medium, the greater the bend toward the normal.
Can you show me with n₁ = 1 and n₂ = 1.5?
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