Five numbers are in HP. The middle term is 1 | vedclass

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(A) Let the five terms in $HP$ be $\frac{1}{a-2d}, \frac{1}{a-d}, \frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}$.Given the middle term is $1$, so $\frac{

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$\frac{11}{2}$

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(A) Let the five terms in $HP$ be $\frac{1}{a-2d}, \frac{1}{a-d}, \frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}$.Given the middle term is $1$, so $\frac{1}{a} = 1$, which implies $a = 1$.The ratio of the second term to the fourth term is $\frac{2}{1}$, so $\frac{\frac{1}{a-d}}{\frac{1}{a+d}} = 2$.This simplifies to $\frac{a+d}{a-d} = 2$, so $a+d = 2a - 2d$, which gives $3d = a$.Since $a = 1$, we have $d = \frac{1}{3}$.The first three terms are $\frac{1}{1-2(\frac{1}{3})}, \frac{1}{1-(\frac{1}{3})}, \frac{1}{1}$.These terms are $\frac{1}{1/3}, \frac{1}{2/3}, 1$, which are $3, \frac{3}{2}, 1$.The sum of the first three terms is $3 + \frac{3}{2} + 1 = 4 + 1.5 = \frac{11}{2}$.

Five numbers are in $HP$. The middle term is $1$ and the ratio of the second and the fourth terms is $2:1$. Then, the sum of the first three terms is

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