multiplication pattern

Question

In a certain geometric sequence, if the first term is 3 and the fifth term is 1875. what is the middle term?

Answer 1

The middle term shall be the geometric mean:

3 : a_n :: a_n : 1875

a_n² = 5625

a_n = 75

 

Answer 2

Hi Young, the geometric sequence starts with a number times a common ratio. The sequence is:

 

ar⁰, ar¹, ar², ar³, ...

 

Note: ar⁰ = a

 

Given: value of third term and fifth term, 3, and 1875.

 

The third term in the sequence is: ar² and the fifth term is: ar⁴.

 

Solution:

 

ar² = 3, therefore, a = 3/r² 

 

ar⁴ = 1875, therefore, a = 1875/r⁴

 

In ar⁴, substitute for a using the a from ar²

 

3/r² * r⁴ = 1875

 

3/r² = 1875

 

r² = 1873 / 3

 

r² = 625

 

r = ±25, keep in mind, geometry sequences are positive.

 

To find the middle term or ar³, substitute for a and r.

 

a = 3/625, from a = 3/r²

 

Then the middle term is:

 

ar³ = 3/625 * 15625

 

ar³ = 3 * 25

 

ar³ = 75

 

Questions?

Answer 3

Here is my set-up from what you wrote: 3, a₂, a₃, a₄, 1875

So we need to find a_2, a₃, a₄, remember that this is

a geometric sequence so a₅=a₁r⁴. We can find the power by finding the difference between 5 and 1: 4. So we have 3r⁴=1875. Divide by 3, r⁴=625. Take the fourth root: r=5. 

From there, the middle three value can be found...

                  a₂=a₁ * 5 = 3 * 5 = 15

                  a₃=a₂ * 5 = 15 * 5 = 75

                  a₄=a₃ * 5 = 75 * 5 = 375