Sow that 1! +2! + 3! + `cdots` + n! cannot be a perfect square for any n `in` N , `n gt=` 4.

If a , a_(1) , a_(2) ,a_(3) , cdots a_(2n), b are in A.P. and a, g_(1) , g_(2) , g_(3) , cdots , g_(2n), b are in G.P. and h is the H.M of a and b then prove that (a_(1)+a_(2n))/(g_(1)g_(2n))+(a_(2)+a_(2n-1))/(g_(2)g_(2n-1))+cdots+ (a_(n)+a_(n+1))/(g_(n)g_(n+1))=(2n)/(h)

Algebraic form of an n^(th) term of an arithmetic sequence is 8n-4. Prove that the sum of the n consecutive terms of this sequence is a perfect square.

Prove that 2^n gt n for all positive integtrs n.

If a_(1) , a_(2), a_(3) , cdots ,a_(n) are in A.P. with a_(1) =3, a_(n) =39 and a_(1) +a_(2) + cdots +a_(n) =210 then the value of n is equal to

Let t_(n) , n = 1,2,3,cdots be the n^(th) term of the A.P. 5,8,11, cdots . Then the value of n for which t_(n) = 305 is

Prove that (1+x)^n ge1+n x , for all natural number 'n', where x gt-1 .

Let {a_(n)} (n gt= 1 ) be a sequence such that a_(1) = 1 and 3a_(n+1)-3a_(n)=1 for all n gt= 1 . Then find the value of a_(2002) .

Prove that (1 + sqrt3i)^n + (1 - sqrt3i)^n = 2^(n+1) cos ((npi)/3) for any positive integer n