Question

Quantos termos tem a PG(1/2, 3/4, ........81/31)?

Answer 1

razão da pg = a2/a1 = (3/4)/(1/2) = 3/4*2 = 3/2
$a_n = a_1* q^{n-1} \\ \\ a_n = \frac{1}{2}*( \frac{3}{2} )^{n-1} \\ \\ \frac{81}{32} = \frac{1}{2}*( \frac{3}{2} )^{n-1} \\ \\ 2*\frac{3^4}{2^5} = (\frac{3}{2})^{n-1} \\ \\ (\frac{3}{2})^4 = (\frac{3}{2})^{n-1} \\ \\ n-1=4 \\ \\ n=5$
Logo
a PG tem 5 termos

Answer 2

a1 = 1/2
a2 = 3/4
an = 81/32

q = a2: a1
q = 3 : 1
      4   2

q = 3 . 2 = 6 (:2) = 3      
     4    1    4 (:2)    2

n = ?

Termo geral da PG:

                n - 1
an = a1. q

               n - 1
81 = 1 . 3
32    2   2 

                   n - 1
162 (:2) =  3
 32  (:2)     2

          n - 1
81 = 3
16    2
   4         n - 1
3      =  3
   4       2
2
    4      n - 1
(3)   = 3
(2)      2

4 = n - 1
4 + 1 = n
5 = n
n = 5
************************
a3 = a2.q

3 . 3 = 9
4   2    8

a4 = a3.q

9 . 3 = 27
8   2    16

a5 = a4.q

27  3 = 81
16  2    32

Resp.: n = 5