\(u_{n+1}=\dfrac{n\left(u_n+2\right)+n^2+1}{n+1}\)

\(\Rightarrow\left(n+1\right)u_{n+1}=nu_n+n^2+2n+1\)

\(\Rightarrow\left(n+1\right)u_{n+1}-\dfrac{1}{3}\left(n+1\right)^3-\dfrac{1}{2}\left(n+1\right)^2-\dfrac{1}{6}\left(n+1\right)=n.u_n-\dfrac{1}{3}n^3-\dfrac{1}{2}n^2-\dfrac{1}{6}n\)

Đặt \(v_n=u.u_n-\dfrac{1}{3}n^3-\dfrac{1}{2}n^2-\dfrac{1}{6}n\Rightarrow\left\{{}\begin{matrix}v_1=1-\dfrac{1}{3}-\dfrac{1}{2}-\dfrac{1}{6}=0\\v_{n+1}=v_n=...=v_1=0\end{matrix}\right.\)

\(\Rightarrow n.u_n-\dfrac{1}{3}n^3-\dfrac{1}{2}n^2-\dfrac{1}{6}n=0\)

\(\Rightarrow u_n=\dfrac{1}{3}n^2+\dfrac{1}{2}n+\dfrac{1}{6}=\dfrac{\left(n+1\right)\left(2n+1\right)}{6}\)

Xét hàm số \(f\left(x\right)=\dfrac{x^{2022}+3x+16}{x^{2021}-x+11}\), ta cần cm

 \(f\left(x\right)\ge x\) (*)

Thật vậy, (*) \(\Leftrightarrow x^{2022}+3x+16\ge x^{2022}-x^2+11x\)

\(\Leftrightarrow x^2-8x+16\ge0\)

 \(\Leftrightarrow\left(x-4\right)^2\ge0\) (luôn đúng)

Vậy \(f\left(x\right)\ge x,\forall x\)

\(\Rightarrow u_{n+1}=f\left(u_n\right)\ge u_n\) nên \(\left(u_n\right)\) là dãy tăng.

Đặt \(u_n+\dfrac{5}{4}=v_n\)

\(GT\Rightarrow\left\{{}\begin{matrix}v_1=\dfrac{9}{4};v_2=\dfrac{13}{4}\\v_{n+2}=2v_{n+1}+3v_n\end{matrix}\right.\)

Ta có CTTQ của dãy \(\left(v_n\right)\) là:

\(v_n=\dfrac{11}{24}.3^n-\dfrac{7}{8}.\left(-1\right)^n\)

(Bạn tự chứng minh theo quy nạp)

\(\Rightarrow u_n=\dfrac{11}{24}.3^n-\dfrac{7}{8}\left(-1\right)^n-\dfrac{5}{4}\) với \(\forall n\in N\text{*}\)

\(\Rightarrow S=2\left(u_1+u_2+...+u_{100}\right)+u_{101}\)

\(=\left[\dfrac{11}{12}\left(3^1+3^2+...+3^{100}\right)-\dfrac{7}{4}\left(-1+1-...+1\right)-\dfrac{5}{2}.100\right]+\dfrac{11}{24}.3^{101}-\dfrac{7}{8}.\left(-1\right)^{101}-\dfrac{5}{4}\)

\(=\dfrac{11}{12}.\dfrac{3^{101}-3}{2}-250+\dfrac{11}{24}.3^{101}+\dfrac{7}{8}\)

\(=\dfrac{11}{24}.\left(2.3^{101}-3\right)-\dfrac{1993}{8}\)

\(=\dfrac{11}{4}.3^{100}-\dfrac{501}{2}\)

\(u_{n+1}=\dfrac{2u_n}{u_n+4}\Leftrightarrow\dfrac{1}{u_{n+1}}=\dfrac{1}{2}+\dfrac{2}{u_n}\)

Đặt \(v_n=\dfrac{1}{u_n}\Rightarrow\left\{{}\begin{matrix}v_1=1\\v_{n+1}=2v_n+\dfrac{1}{2}\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}v_1=1\\v_{n+1}+\dfrac{1}{2}=2\left(v_n+\dfrac{1}{2}\right)\end{matrix}\right.\)

Đặt \(v_n+\dfrac{1}{2}=x_n\Rightarrow\left\{{}\begin{matrix}x_1=\dfrac{3}{2}\\x_{n+1}=2x_n\end{matrix}\right.\)

\(\Rightarrow x_n\) là CSN với công bội 2 \(\Rightarrow x_n=\dfrac{3}{2}.2^{n-1}=3.2^{n-2}\)

\(\Leftrightarrow v_n=x_n-\dfrac{1}{2}=3.2^{n-2}-\dfrac{1}{2}\)

\(\Rightarrow u_n=\dfrac{1}{v_n}=\dfrac{1}{3.2^{n-2}-\dfrac{1}{2}}=\dfrac{2}{3.2^{n-1}-1}\)

Cho \(\left\{{}\begin{matrix}u_1=1\\u_{n+1}=2u_n+6\end{matrix}\right.\)

Tìm số hạng tổng quát của dãy số sau

Với \(n>1\): 

\(n\left(n^2-1\right)u_n=u_1+2u_2+...+\left(n-1\right)u_{n-1}\) (1)

\(\Leftrightarrow n^3-n.u_n=u_1+2u_2+...+\left(n-1\right)u_{n-1}\)

\(\Leftrightarrow n^3.u_n=u_1+2u_2+...+\left(n-1\right)u_{n-1}+n.u_n\) (2)

Thay n bởi \(n-1\) vào (2):

\(\Rightarrow\left(n-1\right)^3u_{n-1}=u_1+2u_2+...+\left(n-1\right)u_{n-1}\) (3)

Từ (1) và (3):

\(\Rightarrow n\left(n^2-1\right)u_n=\left(n-1\right)^2u_{n-1}\)

\(\Leftrightarrow n\left(n+1\right)u_n=\left(n-1\right)^2u_{n-1}\)

\(\Rightarrow u_n=\dfrac{\left(n-1\right)^2}{\left(n+1\right)n}u_{n-1}=\dfrac{\left(n-1\right)^2}{\left(n+1\right)n}.\dfrac{\left(n-2\right)^2}{n\left(n-1\right)}u_{n-2}=...=\dfrac{\left(n-1\right)^2\left(n-2\right)^2....1^2}{\left(n+1\right)n.n\left(n-1\right)...3.2}u_1\)

\(\Rightarrow u_n=\dfrac{\left[\left(n-1\right)!\right]^2}{\dfrac{\left(n+1\right).n^2\left[\left(n-1\right)!\right]^2}{2}}u_1=\dfrac{4}{n^2\left(n+1\right)}\) 

Công thức này chỉ đúng với \(n\ge2\)