Let and be harmonic and geometric sequences respectively. If h 1= g l =(1/2) and h 10= g 10=(1/1024) then

Question

Let < h n > and < g n > be harmonic and geometric sequences respectively. If h 1= g l =(1/2) and h 10= g 10=(1/1024) then (A) h50>g50 (B) h8>g8 (C) displaystyle∑ i =110 h i > displaystyle∑ i =110 g i (D) displaystyle∑ i =1150 h i > displaystyle∑ i =1150 g i

Tardigrade Admin · Accepted Answer

(1/ h 10)=2+9 d =1024 ⇒ d =(1022/9) h n =(1/2+( n -1) (1022)9) ; g n =(1/2 n )=2- n

from the figure it is clear that for n =1 to n =10, h n < g n and for n ≥ 11, h n > g n .

Q. Let $< h_{n} >$ and $< g_{n} >$ be harmonic and geometric sequences respectively. If $h_{1} = g_{l} = \frac{1}{2}$ and $h_{10} = g_{10} = \frac{1}{1024}$ then

$h_{50} > g_{50}$

$h_{8} > g_{8}$

$i = 1 \sum 10 h_{i} > i = 1 \sum 10 g_{i}$

$i = 11 \sum 50 h_{i} > i = 11 \sum 50 g_{i}$

$\frac{1}{h _{10}} = 2 + 9 d = 1024 \Rightarrow d = \frac{1022}{9}$
$h_{n} = \frac{1}{2 + ( n - 1 ) \frac{1022}{9}}; g_{n} = \frac{1}{2 ^{n}} = 2^{- n}$

from the figure it is clear that
for $n = 1$ to $n = 10, h_{n} < g_{n}$ and
for $n \geq 11, h_{n} > g_{n}$ .

Q. Let <hn​> and <gn​> be harmonic and geometric sequences respectively. If h1​=gl​=21​ and h10​=g10​=10241​ then

h50​>g50​

h8​>g8​

i=1∑10​hi​>i=1∑10​gi​

i=11∑50​hi​>i=11∑50​gi​