Question

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

Answer 1

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.