Sequence of series practice · Medium

The arithmetic mean, geometric mean and harmonic mean between two positive real quantities are themselves in

The sum to $\mathrm{n}$ terms of the series $\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\ldots \ldots$ is

The sum of $0.2+0.004+0.00006+0.0000008+\ldots i s$

The sum of three decreasing numbers in A.P. is 27. If $-1,-1,3$ are added to them respectively, the resulting series is in G.P. The numbers are

If 19th term of a non-zero AP is zero, then its (49th term) : (29th term) is

If $\left| a \right| < 1$ and $\left| b \right| < 1,$ then the sum of the series $a\left( {a + b} \right) + {a^2}\left( {{a^2} + {b^2}} \right) + {a^3}\left( {{a^3} + {b^3}} \right) + \ldots$ upto $\infty$, is

If the sum of $1+\frac{1+2}{2}+\frac{1+2+3}{3}+\ldots$ to $n$ terms is $S$, then $S$ is equal to

If $x,y,z$ are in G.P. and $x + 3,y + 3,z + 3$ are in H.P., then

The sum of first 50 terms of $i,-2,-3 i, 4,5 i \ldots$ is

A number consists of three digits in GP. If the sum of extreme digits is greater than twice middle digit by 1 and the sum of the left hand and middle digit is two-third of the sum of the middle and right hand digits, then the number is

If $a,b,c$ are three unequal positive quantities in H.P., then

The arithmetic mean of two numbers is $18 \frac{3}{4}$ and the positive square root of their product is 15 . The larger of the two numbers is

The sum of the infinite series $1 + \frac{1}{{2!}} + \frac{{1.3}}{{4!}} + \frac{{1.3.5}}{{6!}} + \ldots$ is

If $A_1, A_2 ; G_1, G_2$ and $H_1, H_2$ be two A.M.s, G.M.s and H.M.s between two numbers respectively, then $\frac{G_1 G_2}{H_1 H_2} \times \frac{H_1+H_2}{A_1+A_2}=$

If the interior angles of a polygon are in A.P. with common difference $5^\circ$ and the smallest angle $120^\circ$, then the number of sides of the polygon is

If 10 th term of a GP is 9 and 4 th term is 4 , then its 7 th term is

$\frac{{\frac{1}{2}.\frac{2}{2}}}{{{1^3}}} + \frac{{\frac{2}{2}.\frac{3}{2}}}{{{1^3} + {2^3}}} + \frac{{\frac{3}{2}.\frac{4}{2}}}{{{1^3} + {2^3} + {3^3}}} + ... + n$ terms equals

If $\frac{a+b e^y}{a-b e^y}=\frac{b+c e^y}{b-c e^y}=\frac{c+d e^y}{c-d e^y}$, then $a, b, c$ and $d$ are in

If $\mathrm{S}_n$ is the sum of the first $n$ terms of an A.P. and $\mathrm{S}_{3 n}=4 \mathrm{S}_n$, then $\frac{\mathrm{S}_{4 n}}{\mathrm{S}_n}=$

If the sum of four integers in AP is 24 and their product is 945 , then the numbers are

$\frac{1 \times 2^2+2 \times 3^2+3 \times 4^2+----+11(12)^2}{1^2 \times 2+2^2 \times 3+3^2 \times 4+----+11^2(12)}=$?

Sum to n terms of the series $\frac{1}{{1\;.\;\;2\;.\;\;3\;.\;\;4}} + \frac{1}{{2\;.\;\;3\;.\;\;4\;.\;\;5\;}} + \frac{1}{{3\;.\;\;4\;.\;\;5\;.\;6\;}} + \ldots ,$ is

If $k \ne 0$ is a constant and $n \in N$, then, $\lim\limits_{n \to \infty } \,\,\left\{ {\frac{{n!}}{{{{\left( {k\;n} \right)}^n}}}} \right\}$ is equal to

Let $\mathrm{G}$ be the geometric mean of two positive numbers a and $\mathrm{b}$, and $\mathrm{M}$ be the arithmetic mean of $\frac{1}{\mathrm{a}}$ and $\frac{1}{\mathrm{b}}$. If $\frac{1}{\mathrm{M}}: \mathrm{G}$ is $4: 5$, then $a: b$ can be:

The sum of

$\frac{{\frac{1}{2} \cdot \frac{2}{2}}}{{{1^3}}} + \frac{{\frac{2}{2}.\frac{3}{2}}}{{{1^3} + {2^3}}} + \frac{{\frac{3}{2}.\frac{4}{2}}}{{{1^3} + {2^3} + {3^3}}} + ...$ upto $n$ terms is equal to

If $x, y$ and $z$ are in HP, where $z>y>x$. Then, value of $\log (x+z)+\log (x-2 y+z)$ is

The sum of $(x+2)^{n-1}+(x+2)^{n-2}(x+1)+(x+2)^{n-3}$ $(x+1)^2+\ldots+(x+1)^{n-1}$ is equal to

If the AM is twice the GM of the numbers $a$ and $b$, then $a: b$ will be

If $a, b, c$ are in AP and $|a|,|b|,|c|<1$ and $x=1+a+a^2+\ldots+\infty$, $y=1+b+b^2+\ldots+\infty$, $z=1+c+c^2+\ldots+\infty$
Then, $x, y, z$ will be in

Let $S_n$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$, then $S_{14}-S_5$ is :