- 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = ?
- 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 = ?
- 111 + 222 + 333 + 444 + 555 + 666 + 777 + 888 + 999 = ?
- What is going on here? Does this pattern work for 4-digit addends, 5-digit addends, and so on?
Identifying Patterns
Pattern to Formula
Find as many numbers as you can that have the pattern shown here:
52 – 5 = 42 + 4
72 – 7 = 62 + 6
72 – 7 = 62 + 6
Pretty easy, eh? Write a formula to see why.
Egyptian Fractions
Egyptian fractions were typically written in terms of unit fractions, that is, fractions with a numerator of 1. With the exception of a few common, simple fractions like 2/3, non-unit fractions were decomposed into sums of distinct unit fractions.
Examples:
.
Write these fractions as Egyptian fractions:
Sum of Big Fractions
Find this sum without using a calculator. There is a swift way to do it.
- 0
Paths to LOVE
How many paths can you find that spell “LOVE”? You may move vertically and horizontally but not diagonally.
| L | ||||||
| L | O | L | ||||
| L | O | V | O | L | ||
| L | O | V | E | V | O | L |
Difference of Squares 1
Find the difference between 142 and 132. What’s the difference between 1812 and 1802? Write a formula for the difference of the squares of two consecutive integers, in terms of the smaller of the two. What can you say about the number you get each time?
Louie’s Age (Squares)
Louie won’t tell you his age, but it happens that if you square his age this year, and you square his age next year, the difference between those two perfect squares is 75. So how old is he?
Evens Minus Odds
The sum of the first 80 positive odd integers subtracted from the sum of the first 80 positive even integers is:
- 0
- 20
- 40
- 60
- 80