Question

QUANTOS TERMOS DEVEMOS CONSIDERAR NA PG (3,9...) PARA OBTER A SOMA IGUAL A 363?

Answer 1

Soma dos n primeiros termos de uma PG:

$S_n= \frac{a_1(q^n-1)}{q-1}$

Dados:

$S_n=363 \\ q=3\\a_1=3\\n=?$

Cálculo de n:

$S_n= \frac{a_1(q^n-1)}{q-1} \\ \\ 363= \frac{3\,\cdot\,(3^n-1)}{3-1} \\ \\ 363= \frac{3\,\cdot\,(3^n-1)}{2} \\ \\ \frac{363\,\cdot\,2}{3} = 3^n-1 \\ \\ 242=3^n-1 \\ \\ 3^n=243 \\ \\ 3^n=3^5 \\ \\ \boxed{n=5}$

Para que a soma seja 363, o número de termos deve ser n = 5

Answer 2

$PG\{3, 9, ... \}\\\\\\ a_1=3\\ q=9:3=3\\ S_n=363\\n=?\\\\\\ S_n= \dfrac{a_1(q^n-1)}{q-1} \\\\\\ S_n= \dfrac{3(3^n-1)}{3-1} \to~~ 363= \dfrac{3(3^n-1)}{2} \to~~ 726=3(3^n-1)\to\\\\ \dfrac{726}{3}=3^n-1\to~~ 242=3^n-1\to~~242+1=3^n\to~~243=3^n\to\\\\ \not3^5=\not3^n\to ~~ \boxed{n=5}$


R.: 5 termos.