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# RMO 2014 Delhi Region Q.2

Let $$a_1,a_2,\ldots ,a_{2n}$$ be an arithematic progression of positive real numbers with common difference $$d$$ . Let

$\large\left\{\begin{array}{l}\displaystyle\sum^n_{i=1} a^2_{2i-1}=x\\\ \displaystyle\sum^n_{i=1} a^2_{2i}=y\\\ a_n+a_{n+1}=z\end{array}\right.$

Express $$d$$ in terms of $$x, y, z, n$$.

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• You can find the solutions here

Note by Aneesh Kundu
2 years, 5 months ago

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$$\displaystyle d = \frac{y-x}{nz}$$ · 2 years, 5 months ago

got the same answer yesterday in RMO · 2 years, 5 months ago

Hi Sudeep, Any tips for studying coordinate geometry for JEE? · 2 years, 5 months ago

Got the same question in Tamil Nadu's RMO. My approach was as follows:

$y-x=(a^{ 2 }_{ 2n }-a^{ 2 }_{ 1 })+(a^{ 2 }_{ 2n-2 }-a^{ 2 }_{ 3 })+...+(a^{ 2 }_{ 2 }-a^{ 2 }_{ 2n-1 })$

It is a property of an AP that the sum of equidistant terms from the middle term(s) is a constant. Note that $$a_n$$ and $$a_{n+1}$$ are the middle terms of this AP. Hence, their sum $$z$$ is constant for all equidistant terms.

$y-x=(a_{ 2n }+a_{ 1 })(a_{ 2n }-a_{ 1 })+(a_{ 2n-2 }+a_{ 3 })(a_{ 2n-2 }-a_{ 3 })+...+(a_{ 2 }+a_{ 2n-1 })(a_{ 2 }-a_{ 2n-1 })$

$y-x=z(a_{ 2n }-a_{ 1 }) + z(a_{ 2n-2 }-a_{ 3 }) + ... + z(a_{ 2 }-a_{ 2n-1 })$

$y-x=z(a_{ 2n }-a_{ 1 } + a_{ 2n-2 }-a_{ 3 } +a_{ 2 }-a_{ 2n-1 })$

$y-x=z ((a_{2n}-a_{2n-1}) + (a_{2n-2} - a_{2n-3}) + ... + (a_2-a_1))$

The above terms are the difference of two consecutive terms, $$d$$, and there are $$n$$ such terms:

$y-x=znd$

$d=\dfrac{y-x}{zn}$ · 2 years, 5 months ago

In an AP if a1+a5+a10+a15+a25=300, find sum up to 24 terms? · 2 years, 4 months ago

its same as in RMO Karnataka Region!

I got a equation with x,y,z,n,d and d^2.. Couldn't go further! · 2 years, 5 months ago

Did u try using quadratic formula after that? · 2 years, 5 months ago

No! I didn't. May be the equation was wrong, Sudeep Salgia's solution looks right! · 2 years, 5 months ago

See my comment in the discussion. · 2 years, 5 months ago

I think i got it correct. · 2 years, 5 months ago

Well,what is it? How many did you solve in total? · 2 years, 5 months ago

4 · 2 years, 5 months ago

I don't know whether my answer is right or not · 2 years, 5 months ago

(Y- x)/z · 2 years, 5 months ago