Class 11

Math

Algebra

Sequences and Series

If a, b, c are in AP, prove that $a_{2}+c_{2}+4ac=2(ab+bc+ca)$.

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If $a_{n}=43 −(43 )_{2}+(43 )_{3}+...+(−1)_{n−1}(43 )_{n}$ and $b_{n}=1−a_{n}$, then find the least natural number $n_{0}$ such that $b_{n}>a_{n}∀n≥n_{0}$.

If the mean of $5$ observations $x,x+2,x+4,x+6$ and $x+8$ is $11$, find the value of $x$.

Assertion :STATEMENT -1 : If $x,y,z$ are the sides of a triangle such that $x+y+z=1$, then $[32x−1+2y−1+2z−1 ]≥((2x−1)(2y−1)(2z−1))_{1/3}$ Reason :STATEMENT-2 : For positive numbers their A.M., G.M. and H.M. satisfy the relation $A.M.>G.M.>H.M.$

What will be the average of even numbers between $11$ and $63$

Increased by 25%

If A.M of two numbers be twice their G.M then the numbers are in the ratio

If $g_{2}x+g_{2}y≥6,$then the least value of x+y is

Two $A.M.sA_{1}$ and $A_{2}$; two $G.M.sG_{1}$ and $G_{2}$ and two numbers: then $H_{1}+H_{2}$ equals