For , consider the real valued function and . Let and be in an arithmetic progression with mean and positive common difference. If for all , 4 , then the absolute difference between the roots of is...

We have .aligned& f(x)=0 \\& (x-p)^2-q=0 x=p+q, p-qaligned and p=4 a+6 d4 a=(p-32 d) , aligned& |f(a_1)|=500 \\& [\(p-32 d)-p\^2-q]=500 94 d^2-q=500 \\& |(a_1-p)^2-q|=500aligned , aligned& [(a_1-p)^2-q]^2=[(a_2-p)^2-q]^2 \\& (a_1-p)^4+q^2-2 q(a_1-p)^2=(a_2-p)^4+q^2-2 q(a_2-p)^2 \\& (a_1-p)^4-2 q(a_1...

For p,q \in R, consider the real valued function f(x) = (x - p)^{2} - q,x \in R and q > 0. Let a_{1},a_{2},a_{3} and a_{4} be in an arithmetic progression with mean p and positive common difference. If | f( a_{i} ) | = 500 for all i = 1,2,3, 4 , then the absolute difference between the roots of f(x) = 0 is: Sequence & Series (Mathematics)

For

p,q \in R

, consider the real valued function\n

f(x) = (x - p)^{2} - q,x \in R

and

q > 0

. Let\n

a_{1},a_{2},a_{3}

and

a_{4}

be in an arithmetic progression\n with mean

p

and positive common difference. If\n

\left| f\left( a_{i} \right) \right| = 500

for all

i = 1,2,3

,\n 4 , then the absolute difference between the roots of

f(x) = 0

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_{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

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

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_{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?

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

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_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}}

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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