Question

considere a PG de razão em que o primeiro termo é 1 e ultimo é 4096. Quantos termos tem essa PG e qual soma dos teus termos ?

Answer 1

$a_1=1\\ a_n=4096\\ q=4\\ n=?\\\\\\ a_n=a_1\times q ^{n-1}\\\\\\ 4096=1\times 4^{n-1}\to\\\\ 4096= 4^{n-1}\to\\\\ \not4^6=\not4^{n-1}\to\\\\6=n-1\to\\\\ 6+1=n\to\\\\ \boxed{n=7}\\\\\\ PG~\{1,4,16, 64, 256, 1024, 4096\}$



Calculando a Soma:


$S_n= \dfrac{a_1\times (q^n -1)}{q-1} \\ \\ S_n= \dfrac{1\times (4^7 -1)}{4-1}\to \\\\ S_n= \dfrac{1\times (16384-1)}{3}\to \\\\ S_n= \dfrac{1\times 16383}{3}\to \\\\ S_n= \dfrac{ 16383}{3}\to \\\\ \boxed{S_n=5461}$