Progresiones Aritméticas y Geométricas: Fundamentos y Fórmulas

Section role: supports the 'Estructuración: Fundamentos de las Progresiones Aritméticas y Geométricas' section while the teacher explains formal definitions, formulas, and step-by-step procedures on the board. Slides: (1) Title — 'Fundamentos de las Progresiones'; (2) Progresión Aritmética (PA) — bullets: definition as a sequence with a constant difference d, how to calculate d = a_n - a_{n-1}; (3) Fórmulas de la PA — bullets: n-th term formula a_n = a_1 + (n-1)d and sum formula S_n = (n(a_1 + a_n))/2; (4) Ejemplo Práctico PA — bullets: given sequence 5, 8, 11..., find the 20th term and the sum of the first 20 terms with step-by-step resolution; (5) Progresión Geométrica (PG) — bullets: definition as a sequence multiplied by a constant ratio r, how to calculate r = a_n / a_{n-1}; (6) Fórmulas de la PG — bullets: n-th term formula a_n = a_1 * r^(n-1) and sum formula S_n = (a_1(r^n - 1))/(r - 1); (7) Ejemplo Práctico PG — bullets: given sequence 2, 6, 18..., find the 10th term and the sum with step-by-step resolution; (8) Comparación Clave — bullets: contrast between additive growth (PA) and multiplicative growth (PG) connecting back to the initial bacterial and savings examples from the exploration section. Closing slide: '¿Cómo identificas rápidamente si un problema requiere sumar una diferencia o multiplicar por una razón?' — sets up the practical problem-solving section that follows.