The number of common terms in the progressions $4,9,14,19,\dots \dots$, up to $25^{\text{th }}$ term and $3,6,9,12,\dots \dots$, up to $37^{\text{th }}$ term is :

Let $$2^{\text{nd }},8^{\text{th }}$$ and $$44^{\text{th }}$$ terms of a non-constant A. P. be respectively the $$1^{\text{st }},2^{\text{nd }}$$ and $$3^{\text{rd }}$$ terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to -

For $$0(I)If$$$\alpha$ \in(-1,0)$$,then$$bcannot be the geometric mean of $a$ andc$$(II)If$$$\alpha$ \in(0,1)$$,then$$b$$maybethegeometricmeanof$$a$$and$$c$$

The sum of the series $$\frac{1}{1-3\cdot 1^2+1^4}+\frac{2}{1-3\cdot 2^2+2^4}+\frac{3}{1-3\cdot 3^2+3^4}+\dots$$ up to 10 -terms is

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

In an A.P., the sixth term $$a_6=2$$. If the product $$a_1{a}_4{a}_5$$ is the greatest, then the common difference of the A.P. is equal to

If $${\log }_e\mathrm{a},{\log }_e\mathrm{b},{\log }_e\mathrm{c}$$ are in an A.P. and $${\log }_e\mathrm{a}-{\log }_{e}2\mathrm{b},{\log }_{e}2\mathrm{b}-{\log }_{e}3\mathrm{c},{\log }_{e}3\mathrm{c}-{\log }_e$$ a are also in an A.P, then $$a:b:c$$ is equal to

If each term of a geometric progression $$a_1,a_2,a_3,\dots$$ with $$a_1=\frac{1}{8}$$ and $$a_2\ne a_1$$, is the arithmetic mean of the next two terms and $$S_n=a_1+a_2+\dots ..+a_n$$, then $$S_{20}-S_{18}$$ is equal to

Let $$a$$ and $$b$$ be be two distinct positive real numbers. Let $$11^{\text{th }}$$ term of a GP, whose first term is $$a$$ and third term is $$b$$, is equal to $$p^{\text{th }}$$ term of another GP, whose first term is $$a$$ and fifth term is $$b$$. Then $$p$$ is equal to

Let $$S_n$$ denote the sum of first $$n$$ terms of an arithmetic progression. If $$S_{20}=790$$ and $$S_{10}=145$$, then $${\mathrm{S}}_{15}-{\mathrm{S}}_5$$ is :

Let $$a,ar,a{r}^2$$, ............ be an infinite G.P. If \sum_\limits{n=0}^{\infty} a r^n=57 and \sum_\limits{n=0}^{\infty} a^3 r^{3 n}=9747, then $$a+18r$$ is equal to

If the sum of the series $$\frac{1}{1\cdot (1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2\mathrm{d})}+\dots +\frac{1}{(1+9\mathrm{d})(1+10\mathrm{d})}$$ is equal to 5, then $$50\mathrm{d}$$ is equal to :

Let the first three terms 2, p and q, with $$q\ne 2$$, of a G.P. be respectively the $$7^{\text{th }},8^{\text{th }}$$ and $$13^{\text{th }}$$ terms of an A.P. If the $$5^{\text{th }}$$ term of the G.P. is the $$n^{\text{th }}$$ term of the A.P., then $n$ is equal to:

The value of $$\frac{1\times 2^2+2\times 3^2+\dots +100\times (101{)}^2}{1^2\times 2+2^2\times 3+\dots .+100^2\times 101}$$ is

Let three real numbers $$a,b,c$$ be in arithmetic progression and $$a+1,b,c+3$$ be in geometric progression. If $$a>10$$ and the arithmetic mean of $$a,b$$ and $$c$$ is 8, then the cube of the geometric mean of $$a,b$$ and $$c$$ is

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $$\frac{70}{3}$$ and the product of the third and fifth terms is 49. Then the sum of the $$4^{\text{th }},6^{\text{th }}$$ and $$8^{\text{th }}$$ terms is equal to:

If $$\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\dots +\frac{1}{\sqrt{99}+\sqrt{100}}=m$$ and $$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\dots +\frac{1}{99\cdot 100}=\mathrm{n}$$, then the point $$(\mathrm{m},\mathrm{n})$$ lies on the line

For $$x⩾0$$, the least value of $$\mathrm{K}$$, for which $$4^{1+x}+4^{1-x},\frac{\mathrm{K}}{2},16^x+16^{-x}$$ are three consecutive terms of an A.P., is equal to :

Let $$ABC$$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $$ABC$$ and the same process is repeated infinitely many times. If $$\mathrm{P}$$ is the sum of perimeters and $$Q$$ is be the sum of areas of all the triangles formed in this process, then :