Some Special Series — MATHS STD 11 Science — Question
Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSSome Special Series1 Mark
Question
Write the sum to n terms of a series whose rth term is r + 2r
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Answer
Let Tn be the nth term of the series and Sn be the sum to n terms of a series. $\therefore\ \text{T}_\text{r}=\text{r}+2^{\text{r}}$ [given] $\therefore\ \text{S}_\text{n}=\sum\limits^{\text{n}}_{\text{r}=1}(\text{r}+2^{\text{r}})$ $=\sum\limits^{\text{n}}_{\text{r}=1}\text{r}+\sum\limits^{\text{n}}_{\text{r}=1}2^{\text{r}}$ $=\frac{\text{n}(\text{n}+1)}{2}+\big[2^1+2^2\ ...\ 2^{\text{n}}\big]$ $=\frac{\text{n}(\text{n}+1)}{2}+\frac{2(2^\text{n}-1)}{(2-1)}$ $=\frac{\text{n}(\text{n}+1)}{2}+\frac{2\times2^\text{n}-2}{1}$ $=\frac{\text{n}(\text{n}+1)}{2}+2^{\text{n}+1}-2$ $\therefore\ \text{S}_\text{n}=\frac{\text{n}(\text{n}+1)}{2}+2^{\text{n}+1}-2$
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