# Thread: comparison between sum of two sequence

1. ## comparison between sum of two sequence

 in an arithmetic progression and geometric progression the total number of terms are same.the first term and last term of both progressions are also same. if A and G are sum of these expressions then what is relation between the sums.
i tried every way but not able to find rigid proof ?

2. ## Re: comparison between sum of two sequence

I will help you out by providing some notation. The arithmetic progression can be represented by the terms $a_0, \ldots, a_n$ and the geometric progression can be represented by the terms $g_0, \ldots, g_n$. For a progression to be arithmetic, there must be some real number $d$ such that $a_k = a_0 + kd$. For a progression to be geometric, there must be some real number $r$ with $g_k = g_0r^k$. Next, let's figure out the arithmetic and geometric sums:

$A = \sum_{k=0}^n a_k = \sum_{k=0}^n (a_0 + kd) = a_0\sum_{k=0}^n 1 + d\sum_{k=0}^n k$

$G = \sum_{k=0}^n g_k = \sum_{k=0}^n g_0r^k = g_0\sum_{k=0}^n r^k$

3. ## Re: comparison between sum of two sequence

[TD="class: post-message, bgcolor: #F9F9F9"]in an arithmetic progression and geometric progression the total number of terms are same.the first term and last term of both progressions are also same. if A and G are sum of these expressions then what is relation between the sums.[/TD][/TR][TR="class: even, bgcolor: #F9F9F9"][/TR][/TABLE]i tried every way but not able to find rigid proof ?
Originally Posted by SlipEternal
I will help you out by providing some notation. The arithmetic progression can be represented by the terms $a_0, \ldots, a_n$ and the geometric progression can be represented by the terms $g_0, \ldots, g_n$. For a progression to be arithmetic, there must be some real number $d$ such that $a_k = a_0 + kd$. For a progression to be geometric, there must be some real number $r$ with $g_k = g_0r^k$. Next, let's figure out the arithmetic and geometric sums:

$A = \sum_{k=0}^n a_k = \sum_{k=0}^n (a_0 + kd) = a_0\sum_{k=0}^n 1 + d\sum_{k=0}^n k$

$G = \sum_{k=0}^n g_k = \sum_{k=0}^n g_0r^k = g_0\sum_{k=0}^n r^k$
In reply #2 you must realize that the number of terms is $n+1.$

I would be the first to say that I have no idea what the author of this question expects as an answer.
Because both sequences have the same first term, lets call it $a$.

Then the last term being equal we get [TEX]a+nd-=ar^n [TEX]

4. ## Re: comparison between sum of two sequence

Originally Posted by Plato
In reply #2 you must realize that the number of terms is $n+1.$

I would be the first to say that I have no idea what the author of this question expects as an answer.
Because both sequences have the same first term, lets call it $a$.

Then the last term being equal we get [TEX]a+nd-=ar^n [TEX]
Sir we have to prove that sum A is greater than or equal to G

5. ## Re: comparison between sum of two sequence

Solve for $d$ in terms of $a, n$ and $r$. Then evaluate the sums.