1. Tardigrade
  2. Question
  3. Mathematics
  4. Let S= λ mid √(11 λ-2 λ2-8/2 λ) ∈ I , λ>0 . All the elements of set S are the roots of the equation x n + a 1 x n -1+ a 2 x n -2+ ldots ldots+ a n =0 where n denotes the number of elements in S. If ( n-2) is the 3 text rd term of a G.P. such that sum of its infinite term is equal to an then its first term is equal to

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Q. Let $S=\left\{\lambda \mid \sqrt{\frac{11 \lambda-2 \lambda^2-8}{2 \lambda}} \in I , \lambda>0\right\}$. All the elements of set $S$ are the roots of the equation $x ^{ n }+ a _1 x ^{ n -1}+ a _2 x ^{ n -2}+\ldots \ldots+ a _{ n }=0$ where $n$ denotes the number of elements in $S$.
If ( $n-2)$ is the $3^{\text {rd }}$ term of a G.P. such that sum of its infinite term is equal to $a_n$ then its first term is equal to

Sequences and Series

Solution:

Now $n-2=4-2=2=a r^2$...(1)
and $\frac{a}{1-r}=a_4=16$...(2)
(1) and (2) $\Rightarrow r=\frac{1}{2}$ and $a=8$
$\therefore a =8=2 \sqrt{ a _{ n }}=2 \sqrt{ a _4}$