In the prime factorization of `37!`=`2^(a_(1)).3^(a_(2)).5^(a_(3))....37^(a_(n))` the ratio `alpha_(3):alpha_(5)`=

In the prime factorization of 37!= 2^(alpha_(2)).3^(alpha_(3)).5^(alpha_(5))......37^(alpha_(37)) then the ratio alpha_(3):alpha_(5)=

If a_(1),a_(2),a_(3)...,a_(n) are real numbers with a_(n)!=0 and cos alpha+i sin alpha is a root of z^(n)+a_(1)z^(n-1)+a_(2)z^(n-2)+...+a_(n-1)z+a_(n) then the value of a_(1)cos alpha+a_(2)cos2 alpha+a_(3)cos3 alpha+...

(a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=(2a_(2))/(a_(2)+a_(3))

If a_(1),a_(2),".....", are in HP and f_(k)=sum_(r=1)^(n)a_(r)-a_(k) , then 2^(alpha_(1)),2^(alpha_(2)),2^(alpha_(3))2^(alpha_(4)),"....." are in {" where " alpha_(1)=(a_(1))/(f_(1)),alpha_(2)=(a_(2))/(f_(2)),alpha_(3)=(a_(3))/(f_(3)),"....."} .

If a_(1),a_(2),a_(3),......,a_(n) are consecutive terms of an increasing A.P.and (1^(2)-a_(1))+(2^(2)a_(2))+(3^(2)a_(3))+.........+(n^(2)-a_(n))=((n-1)*n*(n+1))/(3) then the value of ((a_(5)+a_(3)-a_(2))/(6)) is equal to

If a_(1),a_(2),a_(3),a_(4),,……, a_(n-1),a_(n) " are distinct non-zero real numbers such that " (a_(1)^(2) + a_(2)^(2) + a_(3)^(2) + …..+ a_(n-1)^(2))x^2 + 2 (a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) + ……+ a_(n-1) a_(n))x + (a_(2)^(2) +a_(3)^(2) + a_(4)^(2) +......+ a_(n)^(2)) le 0 " then " a_(1), a_(2), a_(3) ,....., a_(n-1), a_(n) are in

If a_(1),a_(2),...,a_(n)>0, then prove that (a_(1))/(a_(2))+(a_(2))/(a_(3))+(a_(3))/(a_(4))+...+(a_(n-1))/(a_(n))+(a_(n))/(a_(1))>n

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)

If a_(1),a_(2),a_(3)"....." are in GP with first term a and common rario r, then (a_(1)a_(2))/(a_(1)^(2)-a_(2)^(2))+(a_(2)a_(3))/(a_(2)^(2)-a_(3)^(2))+(a_(3)a_(4))/(a_(3)^(2)-a_(4)^(2))+"....."+(a_(n-1)a_(n))/(a_(n-1)^(2)-a_(n)^(2)) is equal to