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: \(\displaystyle \,S_n\:=\:\frac{n}{2}[4\,+\,2(n\,-\,1)] \:=\:n(n\,+\,1)\)