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1/(a+b), 1/(b+5) and 1/(a+5) are three terms of an arithmetic
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twobagels
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1/(a+b), 1/(b+5) and 1/(a+5) are three terms of an arithmetic [#permalink] New post  19 Feb 2021, 11:09
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\(\frac{1}{(a+b)}\), \(\frac{1}{(b+5)}\) and \(\frac{1}{(a+5)}\) are three terms of an arithmetic progression. Which of the following represents three consecutive terms in the progression in question?

A. 25, 2a, 2b
B. 2a, 25, 2c
C. \(a^2\), \(5^2\), \(b^2\)
D. \(a^2\), \(b^2\), \(5^2\)
E. \(5^2\), \(a^2\), \(b^2\)
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chetan2u
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1/(a+b), 1/(b+5) and 1/(a+5) are three terms of an arithmetic [#permalink] New post  19 Feb 2021, 19:43
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twobagels wrote:
\(\frac{1}{(a+b)}\), \(\frac{1}{(b+5)}\) and \(\frac{1}{(a+5)}\) are three terms of an arithmetic progression. Which of the following represents three consecutive terms in the progression in question?

A. 25, 2a, 2b
B. 2a, 25, 2c
C. \(a^2, 5^2,b^2\)
D. \(a^2,b^2,5^2\)
E. \(5^2,a^2,b^2\)



Start with - Difference in consecutive terms in an AP is the same

\(\frac{1}{(a+b)}, \frac{1}{(b+5)},\frac{1}{(a+5)}\) are in AP.

\(\frac{1}{(a+b)}+\frac{1}{(a+5)}=2*\frac{1}{(b+5)}\)

\((a+5+a+b)(b+5)=2*(a+b)(a+5)\)

\((2a+b+5)(b+5)=(a+b)(2a+10)\)

\(2ab+10a+b^2+10b+25=2a^2+2ab+10a+10b\)

\(b^2+25=2a^2\)

\(25-a^2=a^2-b^2\)

So \(25,a^2,b^2\) are consecutive terms in an AP as the difference between the two consecutive terms is the same : \(25-a^2=a^2-b^2\)

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