If and are distinct real numbers such that then...

aligned& (b) : We have (a^2+b^2+c^2) p^2-2(a b+b c+c d) p+(b^2+c^2+d^2) 0 \\& (a^2 p^2-2 a b p+b^2)+(b^2 p^2-2 b c p+c^2)+(c^2 p^2-2 c d p+d^2) 0 \\& (a p-b)^2+(b p-c)^2+(c p-d)^2 0 \\& (a p-b)^2+(b p-c)^2+(c p-d)^2=0 \\& ( ^2+^2+^2 0 ===0)aligned is possible only when p=ba, p=cb, p=dc ba=cb=dc are ...

If a,b,c,d and p are distinct real numbers such that ( a^{2} + b^{2} + c^{2} )p^{2} - 2(ab + bc + cd)p + ( b^{2} + c^{2} + d^{2} ) \leq 0 then a,b,c,d: Sequence & Series (Mathematics)

a,b,c,d

and

p

are distinct real numbers such that\n

\left( a^{2} + b^{2} + c^{2} \right)p^{2} - 2(ab + bc + cd)p + \left( b^{2} + c^{2} + d^{2} \right) \leq 0

\n then

a,b,c,d

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}}

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

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

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

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

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

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