Let and be harmonic and geometric sequences respectively. If h1=g1=12 and h10=g10=11024 then which of the following is /are TRUE

∑ i = 1 10 h i > ∑ i = 1 10 g i , ∑ i = 11 50 h i > ∑ i = 11 50 g i , h 50 > g 50

Let and be harmonic and geometric sequences respectively. If h1=g1=12 and h10=g10=11024 then which of the following is /are TRUE

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 <img src="data:image/png;base64,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"> it is clear that for n = 1 to n = 10 , h n < g n and for n ≥ 11 , h n > g n

∑ i = 1 10 h i > ∑ i = 1 10 g i

∑ i = 11 50 h i > ∑ i = 11 50 g i

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Let $< h_{n} >$ and $< g_{n} >$ be harmonic and geometric sequences respectively. If $h_{1} = g_{1} = \frac{1}{2} a n d h_{10} = g_{10} = \frac{1}{1024}$ then which of the following is /are TRUE

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$\sum_{i = 1}^{10} h_{i} > \sum_{i = 1}^{10} g_{i}$

$\sum_{i = 11}^{50} h_{i} > \sum_{i = 11}^{50} g_{i}$

$h_{6} < g_{6}$

$h_{50} > g_{50}$

answer is A, B, D.

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Detailed Solution

$\frac{1}{h_{10}} = 2 + 9 d = 1024 \Rightarrow d = \frac{1022}{9}, h_{n} = \frac{1}{2 + (n - 1) \frac{1022}{9}}; g_{n} = \frac{1}{2^{n} = 2^{- n}}$
Question Image it is clear that for $n = 1$ to $n = 10, h_{n} < g_{n}$ and for $n \geq 11, h_{n} > g_{n}$