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If \( a,b,c \) are in G.P then the equation \( ax^2+2bx+c=0 \text { and } dx^2+2ex+f=0 \) have a common root if \( \dfrac{d}{a}, \dfrac{e}{b}, \dfrac{f}{c} \) are in _________
Given, are in GP
So, have equal roots.
Now and the root is
So is the common root.
Now,
are in .
If \( a,\ b,\ c \) are in \( AP \) then______
Given, are in .
Three numbers from an increasing \( G \cdot P \). If the middle term is doubled, then the new numbers are in \( AP \). The common ration of \( GP \) is________
Let three numbers are so
Now, it is given that be in .
(Since )
The sum of \( n \) terms of the series \( \left( \dfrac{1}{1\cdot 2} \right)+\left( \dfrac{1}{2 \cdot 3} \right)+\left( \dfrac{1}{3 \cdot 4} \right)+…\) is________
If \(\dfrac{1}{(b+c)},\dfrac{1}{(c+a)},\dfrac{1}{(a+b)} \) are in AP then________
Given, are in AP.
are in .
The sum of the series \( \dfrac{1}{2!} + \dfrac{1}{4!} + \dfrac{1}{6!} + … \) is________
The third term of the \( GP \) is \( 4 \). The product of the first five terms is _______
Product of first five terms
Let \( T_r \) be the \( r^{th} \) term of an \( AP \), for \( r=1,\ 2,\ 3,\ … \). If for some positive integers \( m,\ n \) we have \( Tm=\dfrac{1}{n}\) and \( Tn=\dfrac{1}{m} \), then \( T_{mn} \) equals______
Let and
(1)
(2)
From above two equations
Also, (from (1))
Now,
The sum of two numbers is \( \dfrac{13}{6} \), an even number of arithmetic means are being inserted between them and their sum exceeds their number by \( 1 \). Then the number of means inserted is ________
Let are two numbers such that
Let be between and .
Then
Given
If the sum of the roots of the quadratic equation \( ax^2+bx+c=0 \) is equal to the sum of the squares of their reciprocals, then \( \dfrac{a}{c},\dfrac{b}{a},\dfrac{c}{b} \) are in _______
Given, Equation is
Let are the roots of this equation.
Now, and
Given,
(Divide on both sides)
are in
are in
are in .
are in
If \( \dfrac{2}{3},k, \dfrac{5}{8} \) are in \( AP \) then the value of \( k \) is _______
are in .
If the sum of the first \( 2n \) terms of the \( \text{AP} \ 2,\ 5,\ 8,\ … \) is equal to the sum of the first \( n \) terms of the \( \text{AP} \ 57,\ 59,\ 61,\ … \), then \( n \) equals________
Given, the sum of the first terms of the the sum of the first terms of the
The sum of \( \text{AP} \ 2,\ 5,\ 8,\ … \) up to \( 50 \) terms is ________
is up to terms first term ,
Common difference
Number of terms
Now, sum
The arithmetric means of first \( n \) odd natural number is _________
Arithmetric means
If the sum of \( 12\text{th} \) and \( 22\text{nd} \) terms of an \( AP \) is \( 100 \), then the sum of the first 33 terms of an \( AP \) is_______
\(\) \begin{aligned}S_{33}&= \dfrac{33}{2}\left[ 2a+(331)d \right]\\&=\dfrac{33}{2}\left[2a+33d \right)\\&=33(a+16d)=33 \times 50=1650 \end{aligned} /latex]
If \( a,\ b,\ c \) are in \(\text{ AP} \) then the value of \( (a+2bc)(2b+ca)(a+2b+c) \) is________
are in
The sets \( S_1,\ S_2,\ S_3\ … \) are given by \( S_1= \left\{ \frac{2}{1} \right\} ,S_2= \left\{ \frac{3}{2} ,\frac{5}{2}\right\} , S_3= \left\{ \frac{4}{3} \frac{7}{3}\frac{10}{3} \right\}, \ … \ , \) then the sum of the numbers in the set \( S_{25} \) is_______
If \( p^{th} \) term of an \( \text{AP} \) is \( q \) and the \( q^{th} \) term is \( p \), then \( 10^{th} \) term is ______
—(1)
—(2)
From (1) [/latex] and (2)
and
Let \( Tr \) be the \( r^{th} \) term of an \(\text{ AP} \) whose first term is \( a \) and common difference is \( d \). If for some positive integers \( m,\ n,\ m \neq n,\ Tm=\dfrac{1}{n} \) and \( Tn=\dfrac{1}{m}, \) then \( ad \) equals_____
— (1)
—(2)
From (1) and (2)
and
Let \( a_1,\ a_2,\ a_3 \ … \) be terms of an \( \text{AP} \). If \( \dfrac{a_1+a_2+…+a_p}{a_1+a_2+…+a_q}=\frac{p^2}{q^2},\ p \neq q \) then \( \dfrac{a_6}{a_{21}} \) equals______
Three number are in \( \text{AP} \) S.t. their sum is \( 18 \) and sum of their squares is \( 158 \). The greatest number among them is _______
Let and are three numbers which are in
(Given)
(Given)
Required numbers are
So greatest number
The sum of the integers from \( 1 \) to \( 100 \) which are divisible by \( 3 \) and \( 5 \) is _____
are the numbers which are divisible by both and
Required sum
The sum of first three terms of a \( \text{GP} \) is \( \dfrac{7}{9} \) and their product is \( \dfrac{8}{27}\). Find the common ratio of the series.
or
The sum of first \( 8 \) terms of the \( \text{GP} \ 2+6+18+54+… \) is_______
For what values of \( x \), the numbers \( 1,\ x,\ \dfrac{3}{4} \) are in \( \text{GP} \)
and are in