Is there a fast way to solve this?

[dragonrider]

2, 12, 72, 432...
"In the sequence above, the first term is 2 and each term after the first is k times the preceding term, where k is a constant. What is the value of the 52nd term divided by the 50th term?"

I know that k is 6. Is there a faster way to get to the 52nd term without having to punch into my calculator 2 x 6 = 12 x 6 = 72 x 6 = 432... all the way to the 52nd term?

 

[Ishuda]

2, 12, 72, 432...
"In the sequence above, the first term is 2 and each term after the first is k times the preceding term, where k is a constant. What is the value of the 52nd term divided by the 50th term?"

I know that k is 6. Is there a faster way to get to the 52nd term without having to punch into my calculator 2 x 6 = 12 x 6 = 72 x 6 = 432... all the way to the 52nd term?

This is called a geometric sequence and each term is given by
x_n = x₀ kⁿ
where x₀ is the initial term (= x₁/k) and n is the term number.

So your sequence is
x_n = \(\displaystyle \frac{2}{3} 6^n\).

So what is \(\displaystyle \frac{x_{52}}{x_{50}}\)?

 

[dragonrider]

This is called a geometric sequence and each term is given by
x_n = x₀ kⁿ
where x₀ is the initial term (= x₁/k) and n is the term number.

So your sequence is
x_n = \(\displaystyle \frac{2}{3} 6^n\).

So what is \(\displaystyle \frac{x_{52}}{x_{50}}\)?

Could you please tell me where you got the 2/3?

 

[Ishuda]

Could you please tell me where you got the 2/3?

The value 6 is called the common ratio, that is the ratio of a term to the previous term. The common ratio here is 6 so the ratio of x1 to x0 is 6, that is
x1/x0 = 6
or
x0 = x1 / 6 = 2 / 6 = 1 / 3

Looks like someone doesn't know how to divide. I won't say who they are but their initials are Ishuda.

 

[dragonrider]

The value 6 is called the common ratio, that is the ratio of a term to the previous term. The common ratio here is 6 so the ratio of x1 to x0 is 6, that is
x1/x0 = 6
or
x0 = x1 / 6 = 2 / 6 = 1 / 3

Looks like someone doesn't know how to divide. I won't say who they are but their initials are Ishuda.

So it shouldn't be 2/3, it should be 1/3?

EDIT: Never mind, I get it now. Thanks!

 

[Ishuda]

So it shouldn't be 2/3, it should be 1/3?

Yep - but that makes no difference to the problem at hand. You could leave it as a generic x₀ and it would cancel out. Another way to get the same answer is
x₅₂/x₅₁ = k
x₅₁/x₅₀ = k
so
x₅₂ = k x₅₁ = k (k x₅₀) = k² x₅₀
so x₅₂/x₅₀ = k²