Mathematical Induction Problem

[mattflint50]

Today when I was studying for my AP exam I came accross a mathematical induction problem that I coult figure out.

{n}= 1 , 2 , 3 , 4 , 5,............
{An}= 2 , 4 , 6 , 8 , 10,...........
{Sn}= 2 , 6 , 12 , 20 , 30,.............


I just cannot seem to find the relationship between {An} and {Sn}. Can you please help me.

 

[daon]

Today when I was studying for my AP exam I came accross a mathematical induction problem that I coult figure out.

{n}= 1 , 2 , 3 , 4 , 5,............
{An}= 2 , 4 , 6 , 8 , 10,...........
{Sn}= 2 , 6 , 12 , 20 , 30,.............


I just cannot seem to find the relationship between {An} and {Sn}. Can you please help me.

Not sure if this is what you are looking for...

$\displaystyle S_1 = A_1 = 2 \\ A_{n} = S_n \, - \, S_{n-1}$

 

[mattflint50]

can you explain what your explanation means ?

Thank you

 

[daon]

can you explain what your explanation means ?

Thank you

Its self explanatory, though I am still unsure if that was the original question, as it wasn't clear.

It says " For n>1 The nth element of the sequence A is equal to the nth element of S minus the (n-1)th element of S. "

 

[soroban]

Hello, mattflint50!

$\displaystyle \;n\;\;\;\,1\;\;\;2\;\;\;3\;\;\;4\;\;\;5\;\;\cdots$
$\displaystyle A_n\;\;\;2\;\;\;4\;\;\;6\;\;\;8\;\;10\;\;\cdots$
$\displaystyle S_n\;\;\;2\;\;\;6\;\;\,12\;\;20\;\;30\;\;\cdots$

I just cannot seem to find the relationship between $\displaystyle A_n$ and $\displaystyle S_n$.

$\displaystyle A_n$ seems to be just $\displaystyle 2n.$

And $\displaystyle S_n$ is the sum of the first $\displaystyle n$ terms.


Since $\displaystyle A_n$ is an arithmetic sequence
$\displaystyle \;\;$with first term $\displaystyle a = 2$ and common difference $\displaystyle d = 2$

the $\displaystyle n^{th}$ term is: $\displaystyle \,A_n\:=\:2\,+\,2(n\,-\,1)\:=\:2n$

and the $\displaystyle n^{th}$ sum is: $\displaystyle \,S_n\:=\:\frac{n}{2}[4\,+\,2(n\,-\,1)] \:=\:n(n\,+\,1)$