THE NTH TERM OF THE SERIES 1+ 1+2/2 + 1+2+3/3 + ... IS n-1/2 n 2 +1/2 n+1/2 n 2 -1/2
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![S_n=1+\frac{1+2}{2}+\frac{1+2+3}{3}+......\\\\T_n=\frac{1+2+3+...+n}{n}=\frac{n+1}{2}\\\\S_n=\Sigma_{n=1}^N\ [ \frac{n+1}{2} ]=\frac{1}{2}[\Sigma\ n+\Sigma\ 1]\\\\=\frac{1}{2}[\frac{n(n+1)}{2} +n]=\frac{n^2+3n}{4}](https://tex.z-dn.net/?f=S_n%3D1%2B%5Cfrac%7B1%2B2%7D%7B2%7D%2B%5Cfrac%7B1%2B2%2B3%7D%7B3%7D%2B......%5C%5C%5C%5CT_n%3D%5Cfrac%7B1%2B2%2B3%2B...%2Bn%7D%7Bn%7D%3D%5Cfrac%7Bn%2B1%7D%7B2%7D%5C%5C%5C%5CS_n%3D%5CSigma_%7Bn%3D1%7D%5EN%5C+%5B+%5Cfrac%7Bn%2B1%7D%7B2%7D+%5D%3D%5Cfrac%7B1%7D%7B2%7D%5B%5CSigma%5C+n%2B%5CSigma%5C+1%5D%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B2%7D%5B%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D+%2Bn%5D%3D%5Cfrac%7Bn%5E2%2B3n%7D%7B4%7D "S_n=1+\frac{1+2}{2}+\frac{1+2+3}{3}+......\\\\T_n=\frac{1+2+3+...+n}{n}=\frac{n+1}{2}\\\\S_n=\Sigma_{n=1}^N\ [ \frac{n+1}{2} ]=\frac{1}{2}[\Sigma\ n+\Sigma\ 1]\\\\=\frac{1}{2}[\frac{n(n+1)}{2} +n]=\frac{n^2+3n}{4}")
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