Expand \(\left( { 2x }^{ 2 }-3\sqrt { 1-{ x }^{ 2 } } \right) ^{ 4 }+({ 2x }^{ 2 }+3\sqrt { 1-{ x }^{ 2 }) } ^{ 4 }\)

Show that the sum of (m + n)^th and (m - n)^th term of an A.P is equal to twice the m^th term.

Using binomial theorem, indicate which of the following two number is larger (1.01)^1000000 (OR)10, 000

Find the last two digits of the number 3⁶⁰⁰

In the binomial expansion of (a+b)ⁿ the coefficients of the 4^th and 13^th terms are equal to each other, find n.

Find the middle term in \({ \left( x-\frac { 1 }{ 2y } \right) }^{ 10 }\)

Find the greatest term in (1 + 2x)⁸ when x = 2.

If H be the H. M. between a and b, then show that (H - 2a) (H - 2b) = H²

Find the n^th term of the series 3 - 6 + 9 -12 + ...

Find the middle term in the expansion of (x +y)⁶.

Find the middle terms in the expansion of (x + y)⁷.

Which two consecutive terms in the expansion (1 +x)¹⁵ have equal coefficients.

Find \(\sum_{k=1}^{n}{1\over k(k+1)}.\)

Find a positive value of m for which the coefficient of x² in the expansion of (1 + x)^m is 6.

Find the \(\sqrt [ 3 ]{ 126 } \) approximately to two decimal places.