RMO 2014 Delhi Region Q.2

Let a1,a2,,a2na_1,a_2,\ldots ,a_{2n} be an arithematic progression of positive real numbers with common difference dd . Let

{i=1na2i12=x i=1na2i2=y an+an+1=z\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 dd in terms of x,y,z,nx, y, z, n.


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Note by Aneesh Kundu
4 years, 10 months ago

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d=yxnz \displaystyle d = \frac{y-x}{nz}

Sudeep Salgia - 4 years, 10 months ago

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got the same answer yesterday in RMO

Souryajit Roy - 4 years, 10 months ago

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Hi Sudeep, Any tips for studying coordinate geometry for JEE?

A Former Brilliant Member - 4 years, 10 months ago

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

yx=(a2n2a12)+(a2n22a32)+...+(a22a2n12)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 ana_n and an+1a_{n+1} are the middle terms of this AP. Hence, their sum zz is constant for all equidistant terms.

yx=(a2n+a1)(a2na1)+(a2n2+a3)(a2n2a3)+...+(a2+a2n1)(a2a2n1)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 })

yx=z(a2na1)+z(a2n2a3)+...+z(a2a2n1)y-x=z(a_{ 2n }-a_{ 1 }) + z(a_{ 2n-2 }-a_{ 3 }) + ... + z(a_{ 2 }-a_{ 2n-1 })

yx=z(a2na1+a2n2a3+a2a2n1)y-x=z(a_{ 2n }-a_{ 1 } + a_{ 2n-2 }-a_{ 3 } +a_{ 2 }-a_{ 2n-1 })

yx=z((a2na2n1)+(a2n2a2n3)+...+(a2a1))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, dd, and there are nn such terms:

yx=zndy-x=znd

d=yxznd=\dfrac{y-x}{zn}

Raj Magesh - 4 years, 10 months ago

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its same as in RMO Karnataka Region!

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

Ranjana Kasangeri - 4 years, 10 months ago

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I think i got it correct.

Aneesh Kundu - 4 years, 10 months ago

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Well,what is it? How many did you solve in total?

Ranjana Kasangeri - 4 years, 10 months ago

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@Ranjana Kasangeri 4

Aneesh Kundu - 4 years, 10 months ago

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@Aneesh Kundu I don't know whether my answer is right or not

Aneesh Kundu - 4 years, 10 months ago

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Did u try using quadratic formula after that?

Aneesh Kundu - 4 years, 10 months ago

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No! I didn't. May be the equation was wrong, Sudeep Salgia's solution looks right!

Ranjana Kasangeri - 4 years, 10 months ago

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@Ranjana Kasangeri See my comment in the discussion.

Aneesh Kundu - 4 years, 10 months ago

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In an AP if a1+a5+a10+a15+a25=300, find sum up to 24 terms?

BHinstitute Harendran B - 4 years, 9 months ago

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(Y- x)/z

prijoe sebastian - 4 years, 10 months ago

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Why?

Aneesh Kundu - 4 years, 10 months ago

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