Question

Determine o número de termos da P.G. (5,10,...,640)

Answer 1

 

 

640 = 5.2^(n-1)

 

2^(n-1) = 640/5

 

2^(n-1) = 128

 

2^(n-1) = 2^7

 

n-1 = 7

  n = 7 + 1  

 

n = 8 

 

 

 

Answer 2

Observe que:

 

$\text{a}_{\text{n}}=\text{a}_1\cdot\text{q}^{\text{n}-1}~~~~(\text{i})$

 

Segundo o enunciado, $\text{a}_{\text{n}}=640$ e $\text{a}_1=5$.

 

Depois disso, note que, $\text{q}=\dfrac{\text{a}_2}{\text{a}_1}=\dfrac{\text{a}_3}{\text{a}_2}=\dots=\dfrac{\text{a}_{\text{n}}}{\text{a}_{\text{n}-1}}=\dfrac{10}{5}=2$

 

Substituindo em $(\text{i})$:

 

$640=5\cdot2^{\text{n}-1}$

 

Dividindo ambos os membros por $5$:

 

$\dfrac{640}{5}=\dfrac{5\cdot2^{\text{n}-1}}{5}$ 

 

$128=2^{\text{n-1}}$

 

Observe que, $128=2^7}$, logo:

 

$2^7=2^{\text{n}-1}$

 

$\text{n}-1=7$

 

$\text{n}=8$

 

Logo, a PG.(5, 10, ... 640) tem $8$ termos.