If the 5^{th} term of a Harmonic Progression | vedclass

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(A) In a Harmonic Progression $(H.P.)$,the terms are the reciprocals of an Arithmetic Progression $(A.P.)$.Let the $A.P.$ have first term $a$ and comm

Vedclass · Accepted Answer

$1/89$

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(A) In a Harmonic Progression $(H.P.)$,the terms are the reciprocals of an Arithmetic Progression $(A.P.)$.Let the $A.P.$ have first term $a$ and common difference $d$.The $5^{th}$ term of $H.P.$ is $1/45$,so the $5^{th}$ term of $A.P.$ is $a + 4d = 45 \dots (i)$.The $11^{th}$ term of $H.P.$ is $1/69$,so the $11^{th}$ term of $A.P.$ is $a + 10d = 69 \dots (ii)$.Subtracting $(i)$ from $(ii)$:$(a + 10d) - (a + 4d) = 69 - 45$$6d = 24 \implies d = 4$.Substituting $d = 4$ into $(i)$:$a + 4(4) = 45 \implies a + 16 = 45 \implies a = 29$.The $16^{th}$ term of the $A.P.$ is $a + 15d = 29 + 15(4) = 29 + 60 = 89$.Therefore,the $16^{th}$ term of the $H.P.$ is $1/89$.

If the $5^{th}$ term of a Harmonic Progression $(H.P.)$ is $1/45$ and the $11^{th}$ term is $1/69$,then its $16^{th}$ term is...

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