Question

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

• A. $h_{50}>g_{50}$
• B. $h_8>g_8$
• C. $\displaystyle\sum_{ i =1}^{10} h _{ i }>\displaystyle\sum_{ i =1}^{10} g _{ i }$
• D. $\displaystyle\sum_{ i =11}^{50} h _{ i }>\displaystyle\sum_{ i =11}^{50} g _{ i }$

Answer 1

Answer: (A), (D)

$ \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 $n \geq 11, h _{ n }> g _{ n }$.