Level – I
1. Prove that the sum of the n arithmetic means inserted between two quantities is n times the single arithmetic mean between them.
2. If x = 1 + a + a2 + a3 + ….. to ∞ (|a| < 1) , y = 1 + b + b2 + b3 + ….. to ∞ (|b| < 1) ,
prove that 1 + ab + a2b2 + a3b3 + …. to ∞ = xy/(x+y-1)
3. Sum up to n terms the series 6 + 66 + 666 +………
4 (i). Sum the series 1.1! + 2.2! + 3.3! + 4.4! + … to n terms.
5. Prove that
$ \displaystyle (x^{n-1} + \frac{1}{x^{n-1}}) + 2(x^{n-2} + \frac{1}{x^{n-2}}) + …. + (n-1)(x+\frac{1}{x}) +n $
$ \displaystyle = \frac{1}{x^{n-1}}(\frac{x^n -1}{x-1})^2 $
6. If a, b, c are positive unequal quantities, then show that
axb – c + bxc – a + cxa – b > a + b + c , (x > 0)
7. Show that the sum of the squares of three consecutive odd numbers increased by 1 is divisible by 12 , but not by 24.
8. If n is a positive integer, prove that 2n > 1+ n√(2n-1)
9 (i). Prove that the greatest value of x2y3z4 , (if x + y + z = 1 , x , y , z > 0) is 210/315
(ii). Find the maximum value of (3 – x)5 (2 + x)4 when x lies between – 2 and 3 .