Question

What is the common ratio of the series `50+40+38...`?

I know how to find the sum of `S_(oo)` when `r` is obvious, like with `sum_(n=1)^(oo)(1/2^n)`, but I don't know how to find the sum when I don't know the common ratio.

Answer 1

The given series is not a Geom. Series, and, as such, we

can not have the Common Ratio.

Explanation:

The Common Ratio, associated with the following

Geometric Series, is `r :`

`a+ar+ar^2+ar^3+...+ar^(n-1)+...," where, "n in NN.`

Its General `n^(th)` Term, `t_n` is, `t_n=ar^(n-1), n in NN.`

We note that, `t_2/t_1=t_3/t_2=...=t_n/t_(n-1)=r.`

As far as the given series is concerned, we find that,

`t_2/t_1=40/50=4/5," whereas, "t_3/t_2=38/40=19/20.`

So, the given series is not a Geom. Series, and, as such, we

can not have the Common Ratio.

Answer 2

The terms fit a quadratic sequence of the form:

`u_n =4n^2-22n+68`

Explanation:

We are only given 3 terms which limits the accuracy of any conclusion drawn. Let us assume that it is a quadratic sequence. Without additional terms we can be certain this is the correct assumption.

Take the difference between consecutive terms, and then take the difference between those terms, as follows:

`{: ("Seq: ",50,,40,,38), ("1st Difference:",,-10,,-2,),("2nd difference:",,,8,,) :}`

So, if this is a quadratic sequence, the terms form a quadratic sequence, of the form:

`{u_n} = an^2 + bx + c`

and the quadratic coefficient is twice the 2nd difference, ie:

`2a=8 => a=4`

We now form a table where we take the original sequence and subtract the quadratic term to give a residue:

`{: (n:, 1,2,3), (n^2:, 1,4,9), ("Seq ("n"): ", 50,40,38), (an^2: ,4,16,36), ("Residue:" ,46,24,2) :}`

So we can now also conclude that the linear part of the sequence (`bx+c`) form the terms `{46,24,2 }`, so we can conclude that `b=-22` and `c=68` and hence the general term of the sequence is:

`\ \ \ \ \ u_n = an^2+bn+c`
`:. u_n =4n^2-22n+68`

Which we can verify as follows:

`{: (n:, 1,2,3), (n^2: ,1,4,9), (ul(" ") ,ul(" "),ul(" "),ul(" ")),(4n^2 ,4,16,36),(-22n ,-22,-44,-66),(+68 ,68,68,68),(ul(" ") ,ul(" "),ul(" "),ul(" ")),(u_n ,50,40,38) :}`