For , let and be one of its root. Then, among the two statements\\ (I) If , then cannot be the geometric mean of and \\ (II) If , then may be the geometric mean of and \\...

aligned& f(x)=(a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 ~b)=0 \\& f(x)=a+b-2 c+b+c-2 a+c+a-2 ~b=0 \\& f(1)=0 \\& 1=c+a-2 ~ba+b-2 c \\& =c+a-2 ~ba+b-2 c \\& If,-1 a+c2aligned, b cannot be G.M. between a and c .aligned& If, 0 c an...

For 0 < c < b < a, let (a + b - 2c)x^{2} + (b + c - 2a)x + (c + a - 2b) = 0 and \alpha eq 1 be one of its root. Then, among the two statements (I) If \alpha \in ( - 1,0), then b cannot be the geometric mean of a and c (II) If \alpha \in (0,1), then b may be the geometric mean of a and c: Sequence & Series (Mathematics)

For

0 < c < b < a

, let

(a + b - 2c)x^{2} + (b + c - 2a)x +

(c + a - 2b) = 0

and

\alpha \neq 1

be one of its root. Then,\n among the two statements\\\n (I) If

\alpha \in ( - 1,0)

, then

b

cannot be the geometric\n mean of

a

and

c

\\\n (II) If

\alpha \in (0,1)

, then

b

may be the geometric mean of\n

a

and

c

Let A_{1},G_{1} and H_{1} denote the arithmetic, geometric and harmonic means . respectively of two distinct positive numbers. For n ? 2 let A_{n - 1} and H_{n - 1} have . arithmetic, geometric and harmonic means as A_{n},G_{n},H_{n} respectively. Which of the following statements is correct?

The sum of the first n terms of the series (1^{2} + 2.2^{2} + 3^{2} + 2.4^{2} + 5^{2} + 2 . 6^{2} + ..) is (\frac{n(n + 1)^{2}}{2}, when n is even. when n is odd, the sum is

For any odd integer n ? 1,n^{3} - (n - 1)^{3} + + ( - 1)^{n - 1}1^{3} =

Let a,b be two non-zero real numbers. If p and r are the roots of the equation x^{2} - 8ax + 2a = 0 and q and s are the roots of the equation \end{enumerate} x^{2} + 12bx + 6b = 0, such that \frac{1}{p},\frac{1}{q},\frac{1}{r},\frac{1}{s} are in A.P., then a^{- 1} - b^{- 1} is equal to

If a_{n} = \frac{3}{4} - ( \frac{3}{4} )^{2} + ( \frac{3}{4} )^{3} + ( - 1)^{n - 1}( \frac{3}{4} )^{n} and b_{n} = 1 - a_{n}, then find the least natural number n_{0} such that b_{n} > a_{n}\forall n ? n_{0}

If a_1, a_2, a_n are in A.P. where a_i>0 for all i show that &\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\cdots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}&=\frac{n-1}{\sqrt{a_1}+\sqrt{a_n}}

If a_{1},a_{2},a_{3}, are in A.P. such that a_{1} + a_{7} + a_{16} = 40, then the sum of the first 15 terms of this A.P. is

Number of 4 -digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to

Different A.P. are constructed with the first term 100 , the last term 199, and integral common differences. The sum of the common differences of all such A.P. having at least 3 terms and at most 33 terms is

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