NDA 2012
Exam NDA
Year 2012
Subject Mathematics
Chapter Sequences and Series NDA 2012 · Mathematics · Sequences and Series
The geometric mean and harmonic mean of two nonnegative...
The geometric mean and harmonic mean of two nonnegative observations are 10 and 8 respectively. Then what is the arithmetic mean of the observations equal to?
Step-by-step solution
a b = 10 a b = 100 2 a b a + b = 8 200 a + b = 8 a + b = 25 ( a − b ) 2 = ( a + b ) 2 − 4 a b = 625 − 400 = 225 a − b = 15 a + b = 25 2 a = 40 a = 20 , b = 5 A.M. = 20 + 5 2 = 12.5. \begin{aligned} \sqrt{a b} & =10 \\ a b & =100 \\ \frac{2 a b}{a+b} & =8 \\ \frac{200}{a+b} & =8 \\ a+b & =25 \\ (a-b)^2 & =(a+b)^2-4 a b \\ & =625-400 \\ & =225 \\ a-b & =15 \\ a+b & =25 \\ 2 a & =40 \\ a & =20, b=5 \\ \text{ A.M. } & =\frac{20+5}{2}=12.5.\end{aligned} ab ab a + b 2 ab a + b 200 a + b ( a − b ) 2 a − b a + b 2 a a A.M. = 10 = 100 = 8 = 8 = 25 = ( a + b ) 2 − 4 ab = 625 − 400 = 225 = 15 = 25 = 40 = 20 , b = 5 = 2 20 + 5 = 12.5.
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