Chapter 9

The sum of the first 20 terms of the series \(1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\ldots\) is? [Online April 16, 2018] (a) \(38+\frac{1}{2^{20}}\) (b) \(39+\frac{1}{2^{19}}\) (c) \(39+\frac{1}{2^{20}}\) (d) \(38+\frac{1}{2^{19}}\)

Let \(a, b, c \in R\). If \(f(x)=a x^{2}+b x+c\) is such that \(a+b+c=3\) and \(\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y})+\mathrm{xy}, \forall \mathrm{x}, \mathrm{y} \in \mathrm{R}\), then \(\sum_{\mathrm{n}=1}^{10} \mathrm{f}(\mathrm{n})\) is equal to: [2017] (a) 255 (b) 330 (c) 165 (d) 190

Let \(\mathrm{S}_{n}=\frac{1}{1^{3}}+\frac{1+2}{1^{3}+2^{3}}+\frac{1+2+3}{1^{3}+2^{3}+3^{3}}+\ldots .\) \(+\frac{1+2+\ldots \ldots+n}{1^{3}+2^{3}+\ldots \ldots .+n^{3}}\), If \(100 \mathrm{~S}_{n}=n\), then \(\mathrm{n}\) is equal to : [OnlineApril 9, 2017] (a) 199 (b) 99 (c) 200 (d) 19

If the sum of the first ten terms of the series \(\left(1 \frac{3}{5}\right)^{2}+\left(2 \frac{2}{5}\right)^{2}+\left(3 \frac{1}{5}\right)^{2}+4^{2}+\left(4 \frac{4}{5}\right)^{2}+\ldots \ldots\), is \(\frac{16}{5} m\) then \(\mathrm{m}\) is equal to : [2016] (a) 100 (b) 99 (c) 102 (d) 101

For \(x \in R, x \neq-1\), if \((1+x)^{2016}+x(1+x)^{2015}+x^{2}\) \((1+x)^{2014}+\ldots .+x^{2016}=\sum_{i=0}^{2016} a_{i} x^{i}\), then \(a_{17}\) is equal to : [Online April 9, 2016] (a) \(\frac{2017 !}{17 ! 2000 !}\) (b) \(\frac{2016 !}{17 ! 1999 !}\) (c) \(\frac{2016 !}{16 !}\) (d) \(\frac{2017 !}{2000 !}\)

If \(\sum_{n=1}^{5} \frac{1}{n(n+1)(n+2)(n+3)}=\frac{k}{3}\), then \(\mathrm{k}\) is equal to |Online April 11, 2015] (a) \(\frac{1}{6}\) (b) \(\frac{17}{105}\) (c) \(\frac{55}{336}\) (d) \(\frac{19}{112}\)

The value of \(\sum_{r=16}^{30}(r+2)(r-3)\) is equal to : [Online April 10, 2015] (a) 7770 (b) 7785 (c) 7775 (d) 7780

\((10)^{9}+2(11)^{1}\left(10^{8}\right)+3(11)^{2}(10)^{7}+\ldots .\) \(+10(11)^{9}=k(10)^{9}\), then \(k\) is equal to: \(\quad\) [2014] (a) 100 (b) 110 (c) \(\frac{121}{10}\) (d) \(\frac{441}{100}\)

The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30 . If the last term exceeds the first term by \(10 \frac{1}{2}\), then the number of terms in the A.P. is: \(\quad\) [Online April 19, 2014] (a) 4 (b) 8 (c) 12 (d) 16

The sum of first 20 terms of the sequence \(0.7,0.77,0.777, \ldots .\) is \(\quad[2013]\) (a) \(\frac{7}{81}\left(179-10^{-20}\right)\) (b) \(\frac{7}{9}\left(99-10^{-20}\right)\) (c) \(\frac{7}{81}\left(179+10^{-20}\right)\) (d) \(\frac{7}{9}\left(99+10^{-20}\right)\)

The value of \(1^{2}+3^{2}+5^{2}+\ldots \ldots \ldots \ldots \ldots \ldots . .+25^{2}\) is: |Online April 25, 2013] (a) 2925 (b) 1469 (c) 1728 (d) 1456

The sum of the series : \(\quad\) [Online April 9, 2013] \(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots \ldots\) upto 10 terms, is : (a) \(\frac{18}{11}\) (b) \(\frac{22}{13}\) (c) \(\frac{20}{11}\) (d) \(\frac{16}{9}\)

Statement-1: The sum of the series \(1+(1+2+4)+\) \((4+6+9)+(9+12+16)+\ldots .+(361+380+400)\) is 8000 . Statement-2: \(\sum_{k=1}^{n}\left(k^{3}-(k-1)^{3}\right)=n^{3}\), for any natural number \(n .\) (a) Statement- 1 is false, Statement- 2 is true. (b) Statement- 1 is true, statement- 2 is true; statement- 2 is a correct explanation for Statement-1. (c) Statement- 1 is true, statement- 2 is true; statement- 2 is not a correct explanation for Statement-1. (d) Statement- 1 is true, statement- 2 is false.

The sum of the series \(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots\) upto 15 terms is \(\quad\) [Online May 12, 2012] (a) 1 (b) 2 (c) 3 (d) 4

The sum of series \(\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\ldots \ldots .\) upto infinity is (a) \(e^{-\frac{1}{2}}\) (b) \(\mathrm{e}^{+\frac{1}{2}}\) (c) \(\mathrm{e}^{-2}\) (d) \(\mathrm{e}^{-1}\)

The sum of series \(\frac{1}{2 !}+\frac{1}{4 !}+\frac{1}{6 !}+\ldots .\) is (a) \(\frac{\left(e^{2}-2\right)}{e}\) (b) \(\frac{(e-1)^{2}}{2 e}\) (c) \(\frac{\left(e^{2}-1\right)}{2 e}\) (d) \(\frac{\left(e^{2}-1\right)}{2}\)

If \(S_{n}=\sum_{r=0}^{n} \frac{1}{{ }^{n} C_{r}}\) and \(t_{n}=\sum_{r=0}^{n} \frac{r}{{ }^{n} C_{r}}\), then \(\frac{t_{n}}{S_{n}}\) is equal to [2004] (a) \(\frac{2 n-1}{2}\) (b) \(\frac{1}{2} n-1\) (c) \(n-1\) (d) \(\frac{1}{2} \eta\)

The sum of the series \(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4} \ldots \ldots \ldots . .\) up to \(\infty\) is equal to (a) \(\log _{e}\left(\frac{4}{e}\right)\) (b) \(2 \log _{e} 2\) (c) \(\log _{e} 2-1\) (d) \(\log _{e} 2\)

\(1^{3}-2^{3}+3^{3}-4^{3}+\ldots+9^{3}=\) (a) 425 (b) \(-425\) (c) 475 (d) \(-475\)

The value of \(2^{1 / 4} .4^{1 / 8} \cdot 8^{1 / 16} \ldots \infty\) is (a) 1 (b) 2 (c) \(3 / 2\) (d) 4