#### 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 + a^{2} + a^{3} + ….. to ∞ (|a| < 1) , y = 1 + b + b^{2} + b^{3} + ….. to ∞ (|b| < 1) ,

prove that 1 + ab + a^{2}b^{2} + a^{3}b^{3} + …. 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

6. If a, b, c are positive unequal quantities, then show that

ax^{b – c} + bx^{c – a} + cx^{a – 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 2^{n} > 1+ n√(2^{n-1})

9 (i). Prove that the greatest value of x^{2}y^{3}z^{4} , (if x + y + z = 1 , x , y , z > 0) is 2^{10}/3^{15}

(ii). Find the maximum value of (3 – x)^{5} (2 + x)^{4} when x lies between – 2 and 3 .