MHB Rudra Rath's question at Yahoo Answers involving an arithmetic series

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    Arithmetic Series
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The discussion centers on solving an arithmetic progression problem posed by Rudra Rath, specifically determining the common difference of an arithmetic series given the first term (a), last term (l), and the sum of the terms (S). The solution involves deriving the common difference (d) using the formulas for the sum of an arithmetic series and substituting values to arrive at the equation d = (l^2 - a^2) / {2S - (l + a)}. The response highlights the relationship between the terms and the sum, confirming the correctness of the derived formula. Additionally, the conversation encourages further inquiries into arithmetic series on a dedicated math forum. This exchange provides a comprehensive understanding of the arithmetic progression concept and its mathematical applications.
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Here is the question:

An arithmetic progression question!?

the first and last terms of an A.P. are a and l respectively. if S be the sum of the terms, then show that the common difference is ( l^2 - a^2 ) / { 2S - (l+a) }

Here is a link to the question:

An arithmetic progression question!? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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Hello Rudra Rath,

We are told:

(1) $$a_1=a$$

(2) $$a_n=a+(n-1)d=\ell$$

(3) $$S=\frac{n(a+\ell)}{2}$$

Now, solving (2) for $d$, we find:

(4) $$d=\frac{\ell-a}{n-1}$$

Solving (3) for $n$, we find:

(5) $$n=\frac{2S}{a+\ell}$$

Substituting for $n$ from (5) into (4), we obtain:

$$d=\frac{\ell-a}{\frac{2S}{a+\ell}-1}\cdot\frac{\ell+a}{\ell+a}=\frac{\ell^2-a^2}{2S-(\ell+a)}$$

Shown as desired.

To Rudra Rath and any other guests viewing this topic, I invite and encourage you to post other arithmetic series questions in our http://www.mathhelpboards.com/f2/ forum.

Best Regards,

Mark.
 
An alternative method:

If you have an arithmetic series, the sum of the first n terms can be found either using [math]\displaystyle S_n = \frac{n}{2} \left( a+ l \right) [/math] or [math]\displaystyle S = \frac{n}{2} \left[ 2a + \left( n - 1 \right) d \right] [/math]. So clearly we can see

[math]\displaystyle \begin{align*} \frac{n}{2} \left( a + l \right) &= \frac{n}{2} \left[ 2a + \left( n - 1 \right) d \right] \\ a + l &= 2a + \left( n - 1 \right) d \\ l - a &= \left( n - 1 \right) d \\ \frac{l - a}{n - 1} &= d \\ \frac{ \left( l - a \right) \left( l + a \right) }{ \left( a + l \right) \left( n - 1 \right) } &= d \\ \frac{ l^2 - a^2}{ n \left( a + l \right) - \left(a + l \right) } &= d \\ \frac{ l^2 - a^2 }{ 2 \left[ \frac{n}{2} (a + l) \right] - (a + l) } &= d \\ \frac{ l^2 - a^2 }{ 2S_n - (a + l) } &= d \end{align*}[/math]
 
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