Question

Can someone explain to me how to do this, please!
This lesson really confused me.
Identify a possible explicit rule for the nth term of the sequence 1, 1/3, 1/5, 1/7, 1/9, ….

Answer 1

The choices are:
\[a _{n}=\frac{ n }{ 2n-1 }\]
\[a _{n}=\frac{ n }{ 2n+1 }\]
\[a _{n}=\frac{ 1 }{ 2n-1 }\]
\[a _{n}=\frac{ 1 }{ 2n+1 }\]

Answer 2

well.. it's asking, on "what pattern are the terms in the sequence show?"

Answer 3

Is the first step to figure out the first differences?

Answer 4

to check what pattern is displaying

Answer 5

Okay, but they aren't constant differences.

Answer 6

First, think about your sequence like this:\[\large\rm \frac{1}{1},~\frac{1}{3},~\frac{1}{5},~\frac{1}{7},...\]If you rewrite the first number like that, it might help you to see what is going on. They are all `odd` denominators.

\(\large\rm 2n-1\) and \(\large\rm 2n+1\) are two ways of writing odd numbers.

Check out \(\large\rm 2n+1\).
If we start counting from n=1,
2(1)+1=3
2(2)+1=5
2(2)+1+7

How bout the other one? \(\large\rm 2n-1\).
Again if we start counting from n=1,
2(1)-1=1
2(2)-1=3
2(3)-1=5

Hmm these second set of numbers seem to match our denominators, yes?

Answer 7

Thank you!