En una progresion geometrica el primer termino es 3 / 4 el ultimo es 2 / 9 y la suma de los terminos es de 65 / 36 ¿de cuantos terminos de compone la progresion ?

A1 = 3 / 4

an = 2 / 9

Sn = 65 / 36

<img src="https://tex.z-dn.net/?f=an%20%3D%20a1%2Ar%5E%7Bn-1%7D%20" />

<img src="https://tex.z-dn.net/?f=Sn%20%3D%20%20%5Cfrac%7B%28an%2Ar-a1%29%7D%7Br-1%7D%20" />

65 / 36 = [(2 / 9)r - (3 / 4)](r - 1)

65(r - 1) = 36[(2 / 9)r - (3 / 4)]

65r - 65 = 8r - 27

65r - 8r = - 27 + 65

57r = 38

r = 38 / 57

r = 2 / 3.

Ahora en <img src="https://tex.z-dn.net/?f=an%20%3D%20a1%2Ar%5E%7Bn-1%7D%20" />

Donde n = Numero de terminos <img src="https://tex.z-dn.net/?f=2%2F9%20%3D%20%283%2F4%29%2A%282%2F3%29%5E%7Bn-1%7D%20" />

<img src="https://tex.z-dn.net/?f=%20%5Cfrac%7B2%2F9%7D%7B3%2F4%7D%3D%282%2F3%29%5E%7Bn-1%7D%20%20" />

<img src="https://tex.z-dn.net/?f=%20%5Cfrac%7B8%7D%7B27%7D%3D%282%2F3%29r%5E%7Bn-1%7D%20%20" />

Aplico logaritmo natural Ln en ambos lados

Ln(8 / 27) = Ln[(2 / 3) ^ (n - 1)]

Ln(8 / 27) = (n - 1)Ln(2 / 3)

n - 1 = [Ln(8 / 27)] / [Ln2 / 3)]

n - 1 = 3

n = 3 + 1

n = 4.

El numero de terminos es 4.

Probemos para n = 4, r = 2 / 3, a1 = 3 / 4

a4 = (3 / 4)(2 / 3) ^ (4 - 1)

a4 = (3 / 4)[(2 / 3) ^ (3)]

a4 = (3 / 4)[8 / 27]

a4 = 2 / 9

Rta : La progresion tiene 4 terminos.