let `f(x)` be a polynomial function of second degree. If `f(1)=f(-1)and a_(1),a_(2),a_(3)` are in AP, then show that `f'(a_(1)),f'(a_(2)),f'(a_(3))` are in AP.

If a_(1)=1,a_(n+1)=(1)/(n+1)a_(n),a ge1 , then prove by induction that a_(n+1)=(1)/((n+1)!)n in N .

If a_(i) gt 0 AA I in N such that prod_(i=1)^(n) a_(i) = 1 , then prove that (a + a_(1)) (1 + a_(2)) (1 + a_(3)) .... (1 + a_(n)) ge 2^(n)

Let a_(1), a_(2),…,a_(n) be fixed real numbers and define a function f(x)=(x-a_(1))(x-a_(2))…(x-a_(n)). What is lim_(xrarra_(1))f(x) ? For some a ne a_(1), a_(2), …..,a_(n) , compute lim_(xrarra)(f(x) .

The number of functions from f:{a_(1),a_(2),...,a_(10)} rarr {b_(1),b_(2),...,b_(5)} is

Find the sum of first 24 terms of the A.P. a_(1) , a_(2), a_(3) ...., if it is know that a_(1) + a_(5) + a_(10) + a_(15) + a_(20) + a_(24) = 225

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

Let b_(i)gt1" for "i=1,2,"......",101 . Suppose log_(e)b_(1),log_(e)b_(2),log_(e)b_(3),"........"log_(e)b_(101) are in Arithmetic Progression (AP) with the common difference log_(e)2 . Suppose a_(1),a_(2),a_(3),"........"a_(101) are in AP. Such that, a_(1)=b_(1) and a_(51)=b_(51) . If t=b_(1)+b_(2)+"........."+b_(51)" and " s=a_(1)+a_(2)+"........."+a_(51) , then

If a_(1),a_(2),a_(3),"........" is an arithmetic progression with common difference 1 and a_(1)+a_(2)+a_(3)+"..."+a_(98)=137 , then find the value of a_(2)+a_(4)+a_(6)+"..."+a_(98) .

If 4a^(2)+9b^(2)+16c^(2)=2(3ab+6bc+4ca) , where a,b,c are non-zero real numbers, then a,b,c are in GP. Statement 2 If (a_(1)-a_(2))^(2)+(a_(2)-a_(3))^(2)+(a_(3)-a_(1))^(2)=0 , then a_(1)=a_(2)=a_(3),AA a_(1),a_(2),a_(3) in R .