Math, asked by anita958, 1 year ago

(a1,a2,.....a4....)is progression where an=n^2/n^3+200 the largest term of progression

Answers

Answered by abhi178
4

Therefore the largest term of progression would be 7.

Given : a₁, a₂, a₃, a₄ ..... is progression where a_n=\frac{n^2}{n^3+200}

To find : The largest term of the progression.

solution : here a_n=\frac{n^2}{n^3+200}

differentiating a_n w.r.t n we get,

\frac{da_n}{dn}=\frac{d\left(\frac{n^2}{n^3+200}\right)}{dn}

using the quotient rule,

\frac{d}{dn}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dn}-u\frac{dv}{dn}}{v^2}

so, \frac{da_n}{dn}=-\frac{n(n^3-400)}{(n^3+200)^2}

here at \frac{da_n}{dn}=0 we get, n = 0, 400⅓

again differentiating w.r.t n we get,

\frac{d^2a_n}{dn^2}=\frac{2(n^6-1400n^3+40000)}{(n^3+200)^3}

at n = 0, we get \frac{d^2a_n}{dn^2} > 0 means at n = 0 a_n will be minimum.

at n = 400⅓ we get \frac{d^2a_n}{dn^2} < 0

means at n = 400⅓ a_n will be maximum.

so the largest term of progression n = 400⅓ ≈ 7

Therefore the largest term of progression would be 7.

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