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Algebra Difficulty 6.3 National olympiad Prove it

【Example 3】Given the general term formula of the sequence as an=32−(−2)na_{n}=3^{2}-(-2)^{n}.
(1) Prove that when kk is an odd number, 1ak+1ak+1<43k+1\frac{1}{a_{k}}+\frac{1}{a_{k+1}}<\frac{4}{3^{k+1}};
(2) Prove that 1a1+1a2+⋯+1an<12(n∈N+)\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}<\frac{1}{2}\left(n \in N^{+}\right).

Solution

(1) When nn is even, we have
( 1 a 1 + 1 a 2 )+ ( 1 a 3 + 1 a 4 )+ + ( 1 a n-1 + 1 a n )0 , then\text{( 1 a 1 + 1 a 2 )+ ( 1 a 3 + 1 a 4 )+ + ( 1 a n-1 + 1 a n )0 , then}1a1+1a2+⋯+1an<1a1+1a2+⋯+1an+1an+1=(1a1+1a2)+(1a3+1a4)+⋯+(1an+1an+1)<432+434+436+⋯+43n+1=12(1−13n+1)<12.\begin{array}{l} \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}<\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}+\frac{1}{a_{n+1}} \\ =\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}\right)+\left(\frac{1}{a_{3}}+\frac{1}{a_{4}}\right)+\cdots+\left(\frac{1}{a_{n}}+\frac{1}{a_{n+1}}\right)<\frac{4}{3^{2}}+ \\ \frac{4}{3^{4}}+\frac{4}{3^{6}}+\cdots+\frac{4}{3^{n+1}}=\frac{1}{2}\left(1-\frac{1}{3^{n+1}}\right)<\frac{1}{2} . \end{array}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.