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Fifth term of a \( \text{GP} \) is \( 2 \), then the product of its \( 9 \) term is ______
The sum of infinity of the progression \( 93+1\dfrac{1}{3}+… \) is ______
which is a .
If \( \left\{ a_n \right\} \) is a sequence with \( a_0=P \) and \( a_na_{n1}=ra_{n1} \) for \( n\geq 1, \) then the terms of the sequence are in ______
Now for
It is in
If \( f(r)=x+\dfrac{1}{2}, \) then the number of real values of \( x \) for which the three unequal terms \( f\left( x \right), f\left( 2x \right)\ \text{ and }\ f\left( 4x \right) \) are in \( HP \) is_________
Since, are in , then must be in
So, there is one real value of
The sum of \( 0.2+0.22+0.222+… \) to \( n \) term is equal to _______
up to terms
Suppose \( a,\ b\ \text{ and }\ c \) are in \( AP\ \text{ and }\ a^2,\ b^2,\ c^2 \) are in \( GP \). If \( a<b<c \) and \( a+b+c=\dfrac{3}{2} \), then the value of \( a \) is______
If the \( \left( p+q \right)^{th} \) term of a \( \text{GP} \) is \( m \) and the \( \left( p+q \right)^{th} \) term is \( n \), then the \( p^{th} \) term is ______
If \( 0 \) < \( \theta, \dfrac{\pi}{2}, x\sum_{n=0}^{\infty} \cos^n \theta, y=\sum_{n=0}^{\infty} \sin^n \theta \text { and } z=\sum_{n=0}^{\infty} \cos^n \theta \sin^n \theta \) then______
Similarly we can obtain
and
The product \( \left( 32 \right) \left( 32 \right)^{\frac{1}{6}}\left( 32 \right)^{\frac{1}{36}}… \) to \( \infty \) is_______
If \( S \) is the sum of an infinite \( GP \text { and } a \) is the first term, then the common ratio \( r \) is given by______
For an infinite
Sum
If \( H \) is the \( \text{HM} \) between \( p \text{ and } q \), then the value of \( \dfrac{H}{P}+\dfrac{H}{Q} \) is______
(i)
(ii)
If \( x,\ y,\ z \) are in \( HP \), them \( log\left( x+z \right)+log\left( x2y+z \right) \) is equal to_____
If \( \log_x ax, \log_x bx,\ \text{and }\log_xcx \) are in \( \text{AP} \), where \( a,\ b,\ c \) and \( x \) belong to \( \left( 1, \infty \right), \) then \( a,\ b,\ \text{ and}\ c \) are in_____
As
If \( x,y, \text{ and } z\) are in \( GP \), then \( \log_x 10,\log_y 10\ \text{ and }\ log_z 10 \) are in_____
As are in GP.
\( GM \text{ and } HM \) of two numbers are \( 10 \text{ and } 8 \), respectively, then the numbers are_______
The numbers are
If \( a,\ b,\c \) are in \( GP \) and \( x,y \) are \( \text {A.M} \) of \( a, b \) and \( b,c \) respectively, then \( \dfrac{1}{x} +\dfrac{1}{y} \) is equal to______
If \( a \) is positive and also \( A,G \) are the \( \text{AM} \) and the \( \text{GM} \) of the roots of \( x^22ax+a^2=0 \) respectively, then_______
Now
and
Three numbers whose sum is \( 15 \) are in \( \text{ AP} \) If they are added by \( 1,4 \text{ and } 19 \) respectively, then they are in \( \text{ GP} \). The numbers are______
Let three numbers of be
Also
If \( \text{AM} \) of two numbers is twice of their \( \texT{GM} \), then the ratio of greatest number to smaller number is______
Let two numbers are , such that
–(1)
Now
—(2)
From (1) and (2) we get,
Required ratio
If \( H_1 \) and \( H_2 \) are two \( \text{HM} \) between two positive numbers \( a \text { and } b \) and \( (a\neq b) \), and \( A \text { and }G \) are the \( \text{AM} \) and \( \text{GM} \) between \( a \) and \( b \) respectively, then \( \dfrac{H_2+H_1}{H_2H_1} \) is______
are in HP.
and
–(i)
–(ii)
–(iii)
From (i), (ii) and (iii) we get
If \( A_1,\ A_2,\ G_1,\ G_2 \) and \( H_1,\ H_2 \) are two \( AM’s,\ GM’s \) and \( HM’s \) between two quantities, then the value of \( \dfrac{G_1G_2}{H_1H_2} \) is_____
are in AP.
–(i)
are in GP.
–(ii)
are in HP.
The value of \( 2^{\frac{1}{4}}\cdot 4^{\frac{1}{8}}\cdot 8^{\frac{1}{16}}…\infty \) is______
Sum of the series \( 1+\dfrac{4}{5}+\dfrac{7}{5^2}+\dfrac{10}{5^3}+…\infty \) is________
, if <
and
\( \sum_{n=8}^{17}\frac{1}{\left( n+2 \right)\left( n+3 \right)} \) is equal to_______
\( \sum_{k=1}^{2n+1}\left( 1 \right)^{k1}K^2 \) equals_____
–(i)