If 0 < \theta ,\phi < {\pi \over 2},x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta } ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi } and $$z=\sum _{n=0}^{\infty }{\cos }^{2n}\theta .{\sin }^{2n}\varphi$$ then :

The minimum value of $$f(x)=a^{a^x}+a^{1-a^x}$$, where a, $$x\in R$$ and a > 0, is equal to :

The sum of the infinite series 1 + {2 \over 3} + {7 \over {{3^2}}} + {{12} \over {{3^3}}} + {{17} \over {{3^4}}} + {{22} \over {{3^5}}} + ...... is equal to :

In an increasing geometric series, the sum of the second and the sixth term is {{25} \over 2} and the product of the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to :

The sum of the series \sum\limits_{n = 1}^\infty {{{{n^2} + 6n + 10} \over {(2n + 1)!}}} is equal to :

If $$\alpha$$, $$\beta$$ are natural numbers such that 100$$\alpha$$ $$-$$ 199$$\beta$$ = (100)(100) + (99)(101) + (98)(102) + ...... + (1)(199), then the slope of the line passing through ($$\alpha$$, $$\beta$$) and origin is :

{1 \over {{3^2} - 1}} + {1 \over {{5^2} - 1}} + {1 \over {{7^2} - 1}} + .... + {1 \over {{{(201)}^2} - 1}} is equal to

Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithmetic progression. If (S2 $$-$$ S1) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :

If sum of the first 21 terms of the series $${\log }_{9^{1/2}}x+{\log }_{9^{1/3}}x+{\log }_{9^{1/4}}x+.......$$, where x > 0 is 504, then x is equal to

Let Sn denote the sum of first n-terms of an arithmetic progression. If S10 = 530, S5 = 140, then S20 $$-$$ S6 is equal to:

Let Sn be the sum of the first n terms of an arithmetic progression. If S3n = 3S2n, then the value of {{{S_{4n}}} \over {{S_{2n}}}} is :

The sum of the series {1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}} when x = 2 is :

If the sum of an infinite GP a, ar, ar2, ar3, ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, ....... is :

If 0 < x < 1, then {3 \over 2}{x^2} + {5 \over 3}{x^3} + {7 \over 4}{x^4} + ....., is equal to :

If for x, y $$\in$$ R, x > 0, y = log10x + log10x1/3 + log10x1/9 + ...... upto $$\infty$$ terms and {{2 + 4 + 6 + .... + 2y} \over {3 + 6 + 9 + ..... + 3y}} = {4 \over {{{\log }_{10}}x}}, then the ordered pair (x, y) is equal to :

If 0 < x < 1 and y = {1 \over 2}{x^2} + {2 \over 3}{x^3} + {3 \over 4}{x^4} + ...., then the value of e1 + y at x = {1 \over 2} is :

The sum of 10 terms of the series {3 \over {{1^2} \times {2^2}}} + {5 \over {{2^2} \times {3^2}}} + {7 \over {{3^2} \times {4^2}}} + .... is :

Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3 r2, then r2 $$-$$ d is equal to :

Let a1, a2, a3, ..... be an A.P. If {{{a_1} + {a_2} + .... + {a_{10}}} \over {{a_1} + {a_2} + .... + {a_p}}} = {{100} \over {{p^2}}}, p $$\ne$$ 10, then {{{a_{11}}} \over {{a_{10}}}} is equal to :

Let Sn = 1 . (n $$-$$ 1) + 2 . (n $$-$$ 2) + 3 . (n $$-$$ 3) + ..... + (n $$-$$ 1) . 1, n $$\ge$$ 4. The sum \sum\limits_{n = 4}^\infty {\left( {{{2{S_n}} \over {n!}} - {1 \over {(n - 2)!}}} \right)} is equal to :