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makieldeoc
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Tyro
Asked: 2021-03-30T02:43:44+00:00 In: ,

harmonic sequence

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Find the 6th term of harmonic sequence whose 1st term is 3 and 3rd term is 3/5.

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  2. acedstud
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    The nth term of a harmonic sequence is given by T_n=\dfrac{1}{a+(n-1)d}.

    Since the first term is 3, thus \dfrac{1}{a}=3, and the third term is \dfrac{3}{5}, thus \dfrac{1}{a+2d}=\dfrac{3}{5}. Taking the reciprocal gives that a=\dfrac{1}{3} and a+2d=\dfrac{5}{3}\Rightarrow\dfrac{1}{3}+2d=\dfrac{5}{3}\Rightarrow2d=\dfrac{5}{3}-\dfrac{1}{3}=\dfrac{4}{3}.

    \Rightarrow d=\dfrac{2}{3}

    Therefore, the 6th term is \dfrac{1}{a+5d}=\dfrac{1}{\frac{1}{3}+5(\frac{2}{3})}=\dfrac{1}{\frac{1}{3}+\frac{10}{3}}=\dfrac{1}{\frac{11}{3}}=\dfrac{3}{11}.

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