Question

13．已知數(shù)列{a_n}為等比數(shù)列，且${a_{2015}}+{a_{2017}}=\int_0^2{\sqrt{4-{x^2}}}dx$，則a₂₀₁₆（a₂₀₁₄+a₂₀₁₈）的最小值為$\frac{{π}^{2}}{2}$．

Answer 1

分析 根據(jù)定積分的幾何意義先求出a₂₀₁₅+a₂₀₁₇=π，再根據(jù)等比數(shù)列的性質(zhì)和基本不等式即可求出．

解答 解：${∫}_{0}^{2}$$\sqrt{4-{x}^{2}}$dx表示以原點(diǎn)為圓心以2為半徑的圓的面積的四分之一，${∫}_{0}^{2}$$\sqrt{4-{x}^{2}}$dx=π，
∴${a_{2015}}+{a_{2017}}=\int_0^2{\sqrt{4-{x^2}}}dx$=π，
∴a₂₀₁₆（a₂₀₁₄+a₂₀₁₈）=a₂₀₁₆•a₂₀₁₄+a₂₀₁₆•a₂₀₁₈=a₂₀₁₅²+a₂₀₁₇²≥$\frac{1}{2}$（a₂₀₁₅+a₂₀₁₇）²=$\frac{{π}^{2}}{2}$，
故答案為：$\frac{π^2}{2}$．

點(diǎn)評(píng) 本題考查等比數(shù)列的性質(zhì)，利用數(shù)形結(jié)合的思想，涉及定積分的求解，屬中檔題．