1. lim √3n^2 +1 +n/1-2n 2.Tổng 1 CSN lùi vô hạn là 5/3 tổng 3 số hạng đầu là 39/25.Tìm u1,q

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Đáp án:

$1)-\dfrac{1+\sqrt{3}}{2}\\ 2)q=\dfrac{2}{5}; u_1=1.$

Giải thích các bước giải:

$1)\\ \lim \dfrac{\sqrt{3n^2+1}+n}{1-2n}\\ =\lim \dfrac{\sqrt{3+\dfrac{1}{n^2}}+1}{\dfrac{1}{n}-2}\\ = \dfrac{\sqrt{3}+1}{-2}\\ =-\dfrac{1+\sqrt{3}}{2}\\ 2)$

Tổng $3$ số hạng đầu là $\dfrac{39}{25}$

$\Rightarrow u_1+u_1q+u_1q^2=\dfrac{39}{25}(1)$

Tổng CSN lùi vô hạn này bằng $\dfrac{5}{3}$

$\Rightarrow \dfrac{u_1}{1-q}= \dfrac{5}{3}\\ \Rightarrow u_1=\dfrac{5(1-q)}{3}$

Thế vào $(1) \Rightarrow \dfrac{5(1-q)}{3}+\dfrac{5(1-q)}{3}.q+\dfrac{5(1-q)}{3}.q^2=\dfrac{39}{25}$

$\Leftrightarrow (1-q)+(1-q)q+(1-q)q^2=\dfrac{117}{125}\\ \Leftrightarrow (1-q)(1+q+q^2)=\dfrac{117}{125}\\ \Leftrightarrow 1 - q^3=\dfrac{117}{125}\\ \Leftrightarrow q^3=\dfrac{8}{125}\\ \Leftrightarrow q=\dfrac{2}{5}\\ u_1=\dfrac{5(1-q)}{3}=1.$

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