Solve for nGeometric Sequences: Solving for Number of Terms

[nomad2817]

Homework Statement

Hi, I was trying to work out this question, but i kinda got stuck. Can anyone help me please?
Thanks

4. Find the number of terms in each of these geometric sequences.

2,10,50...1250

Homework Equations

ar^n-1

The Attempt at a Solution

1250=2x5^n-1

 

[Cyosis]

I take it you mean $1250=2*5^{n-1}$ with n starting at 1? You're on the right track. What is the problem with the expression you found so far? You are unable to solve it for n? Hint: logarithm

 

[nomad2817]

i don't get what the next step is

 

[Cyosis]

What does n represent in that formula?

 

[nomad2817]

n is the term, but i don't know how to find it

 

[Cyosis]

"The term", you mean is the number of terms in the sequence. The first step in solving it is to put 5^(n-1) on one side and the other terms on the other side. Then take the logarithm on both sides.

 

[nomad2817]

so do i have to balance both sides and eliminate 1250?

 

[Cyosis]

An equation is balanced by definition. If you substitute x=5^(n-1) then the equation becomes 1250=2x. Now solve this equation for x and then back substitute x.

ps. Are you familiar with logarithms?

 

[nomad2817]

Thank you very much. I've just started learning about it, so I'm trying to get the hang of it, but thank you for helping me.

 

[Cyosis]

You're welcome, but have you found the solution?

 

[nomad2817]

1250= 2x so x=1250/2 x=625

oh no, I'm getting confused again. sorry

 

[Cyosis]

Correct x=625 and because we substituted x=5^(n-1) we know that 5^(n-1)=625. We now want to write 625 in terms of 5 raised to a certain exponent. For example we can write 8=2^3, 125=5^3. Try to write 625 in a similar way.

 

[nomad2817]

so it can be 625=25^2

 

[Cyosis]

Yes that is correct although not entirely what I had in mind. Note that you can write 25=5^2 therefore 25^2=(5^2)^2=5^4=625.

So now your equation becomes:

$$5^{n-1}=5^4$$

Therefore n-1 equals...?

 

[nomad2817]

n-1= 5

 

[Cyosis]

No the two exponents need to be equal to each other, it is the only way the equation can hold. So I will ask you again n-1=?

What you're saying now is that 5^5=5^4. Convince yourself that this cannot be correct.

 

[nomad2817]

ohhhh n-1= 5-1?

 

[Cyosis]

Why do you think so, explain your reasoning behind it.

 

[nomad2817]

because 5^(5-1) which equals to 5^4

 

[Cyosis]

Well your reasoning and your answer are correct although a bit of a detour. You know that both exponents need to be the same therefore n-1=4.

 

[nomad2817]

ohhhhh, i see now. Thank you