Problemas matemáticos e raciocínio lógico · High School Algebra (Sequences & Series)
Visualizador de progressões aritméticas e geométricas
Visualizador de progressões aritméticas (PA) e geométricas (PG): soma de Gauss e xadrez
- No navegador
- Sem cadastro
- Dedução ao vivo
- Modo projetor
- Folha de exercícios A4
Janela de demonstração matemática● 60 FPS
Variáveis de controle do experimento
células
Deduções e métricas em tempo real
Soma da série S_n
--
Último termo a_n
--
Pares de Gauss
--
Ordem de grandeza
--
Dedução de fórmulas em tempo real
Fórmula fundamental:
--Substituição de valores:
--Conclusão: --
Exemplos clássicos e armadilhas de exame
Worked Problem
Problema: Calculate 2 + 4 + 6 + ... + 100. Show term count and sum.
Derivação passo a passo:
- a1=2, an=100, d=2. Terms n = (100 - 2)/2 + 1 = 50. Sum S = (2 + 100)*50 / 2 = 102 * 25 = 2550.
Worked Problem
Problema: Deposit 1 cent on day 1, 2 cents on day 2, doubling daily for 30 days. Find day 30 deposit and total sum.
Derivação passo a passo:
- Day 30 = 2²⁹ cents ≈ $5.37 million. Total = 2³⁰ - 1 cents ≈ $10.74 million dollars.
Folha de investigação em aula e exercícios
Q1. Variant 1: Pile of logs: top row 4, bottom row 12, each row adds 1 log. Find rows and total logs.
Answer Key: Rows n = (12 - 4)/1 + 1 = 9 rows. Total logs S = (4 + 12)*9 / 2 = 72 logs.
Q2. Variant 2: Sum of first 20 odd numbers? Sum of all odd numbers from 1 to 99?
Answer Key: Sum of first n odds is n². 1) First 20 odds = 20² = 400. 2) 1 to 99 has 50 terms: sum = 50² = 2500.
Q3. Variant 3: 0.1mm paper folded 30 times. How thick? Does it exceed Mt. Everest?
Answer Key: Thickness = 0.1mm * 2³⁰ ≈ 107.4 km (107,374 meters), which is 12 times taller than Mt. Everest (8848m).
Perguntas frequentes e erros comuns
How did 10-year-old Gauss sum 1 to 100 in mere seconds?
He spotted symmetric pairs: 1+100=101, 2+99=101, etc. 50 identical pairs of 101 = 50 * 101 = 5050, astounding his teacher.
How to remember arithmetic sequence nth term and sum formulas?
nth term: an = a1 + (n - 1)d. Sum: Sn = n(a1 + an)/2 (identical to trapezoid area formula [top + bottom]*height / 2).
Why was the King terrified after granting the vizier's chessboard wheat request?
The exponential series 2^64 - 1 yields 18.4 quintillion grains (~140 billion tons), exceeding worldwide wheat production for centuries.
How many paper folds to reach the Moon (384,000 km)?
Only 42 folds! 0.1mm * 2⁴² ≈ 440,000 km, surpassing the distance to the Moon. This illustrates the exponential power of compounding.