Nth Term of A.G.P

Q.

The sum of $4$ numbers in GP is $60$ and the AM of the first and last numbers is $18$. Find the first term and the common difference of the GP.

Q. The sum of the series $\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+\frac{x^4}{1-x^8}+......$ to infinite terms, if |x| > 1 is

1. 
2. 
3. 
4. 

Q.

Which is the first negative term of the AP $20,\frac{77}{4},\frac{37}{2},\frac{71}{4}$.

Q. The sum of the series $1+2x+3x^2+4x^3+.....$ upto n terms is

1. $\frac{1-(n+1)x^n+n{x}^{n+1}}{(1-x{)}^2}$
2. $\frac{1-x^n}{(1-x}$
3. $x^{n+1}$
4. None of these

Q. The sum of $1+\frac{2}{5}+\frac{3}{5^2}+\frac{4}{5^3}+.......$ upto n terms is
[MP PET 1982]

1. $\frac{25}{16}-\frac{4n+5}{16\times 5^{n-1}}.$
2. $\frac{3}{4}-\frac{2n+5}{16\times 5^{n+1}}.$
3. $\frac{1}{2}-\frac{5n+1}{3\times 5^{n+2}}.$
4. $\frac{3}{7}-\frac{3n+5}{16\times 5^{n-1}}.$

Q.

The sum of series $\sum _{x=1}^n\log \left(\frac{2^x}{3^{x-1}}\right)$ is

1. $\log {\left(\frac{2^{n-1}}{3^{n+1}}\right)}^{\frac{n}{2}}$

2. $\log {\left(\frac{2^{n+1}}{3^{n-1}}\right)}^{\frac{n}{2}}$

3. $\log {\left(\frac{3^{n-1}}{2^{n+1}}\right)}^{\frac{n}{2}}$

4. $\log {\left(\frac{3^{n+1}}{2^{n-1}}\right)}^{\frac{n}{2}}$

Q. The sum of the first n terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+.....$ is
[IIT 1988; MP PET 1996; RPET 1996, 2000; Pb. CET 1994; DCE 1995, 96]

1. $2^n-n-1$
2. $1-2^{-n}$
3. $n+2^{-n}-1$
4. $2^n-1$

Q. The sum of the first n terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+.....$ is
[IIT 1988; MP PET 1996; RPET 1996, 2000; Pb. CET 1994; DCE 1995, 96]

1. $1-2^{-n}$
2. $2^n-n-1$
3. $n+2^{-n}-1$
4. $2^n-1$

Q. The sum of the series $1+2x+3x^2+4x^3+.....$ upto n terms is

1. $\frac{1-(n+1)x^n+n{x}^{n+1}}{(1-x{)}^2}$
2. $\frac{1-x^n}{(1-x}$
3. $x^{n+1}$
4. None of these

Q. The sum of $1+\frac{2}{5}+\frac{3}{5^2}+\frac{4}{5^3}+.......$ upto n terms is
[MP PET 1982]

1. $\frac{25}{16}-\frac{4n+5}{16\times 5^{n-1}}.$
2. $\frac{3}{4}-\frac{2n+5}{16\times 5^{n+1}}.$
3. $\frac{3}{7}-\frac{3n+5}{16\times 5^{n-1}}.$
4. $\frac{1}{2}-\frac{5n+1}{3\times 5^{n+2}}.$

Q. The sum of $1+\frac{2}{5}+\frac{3}{5^2}+\frac{4}{5^3}+.......$ upto n terms is
[MP PET 1982]

1. $\frac{25}{16}-\frac{4n+5}{16\times 5^{n-1}}.$
2. $\frac{3}{4}-\frac{2n+5}{16\times 5^{n+1}}.$
3. $\frac{3}{7}-\frac{3n+5}{16\times 5^{n-1}}.$
4. $\frac{1}{2}-\frac{5n+1}{3\times 5^{n+2}}.$

Q. If $S_1,S_2,S_3$ are the sums of the terms of n three series in A.P., the first term of each being 1 and the respectively common difference being 1, 2, 3; prove the
$S_1+S_3=2S_2$

Q. The sum of the first n terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+.....$ is
[IIT 1988; MP PET 1996; RPET 1996, 2000; Pb. CET 1994; DCE 1995, 96]

1. $2^n-n-1$
2. $1-2^{-n}$
3. $n+2^{-n}-1$
4. $2^n-1$