If the roots of the equation (q - r) x² + (r - p)x + p - q = 0 are equal, then show that p, q and r are in A.P.

If a, b, c are respectively the p^th q^th and r^th terms of a GP. show that (q - r) log a + (r - p) log b + (p - q) log c = 0.

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

Find the sum up to the 17^th term of the series \(\frac { { 1 }^{ 3 } }{ 1 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 } }{ 1+3 } +...+\frac { { 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 } }{ 1+3+5 } +......\)

Compute the sum of first n terms of the following series 8 + 88 + 888 + .......

Compute the sum of first n terms of 1 + (1 + 4) + (1 + 4 + 4²) + (1 + 4 + 4² + 4³) + ...

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2^nd hour, 4^th hour and n^th hour?

Find the Constant term of \(\left( { 2x }^{ 3 }-\frac { 1 }{ { 3x }^{ 2 } } \right) ^{ 5 }\)

Find \(\sqrt [ 3 ]{ 1001 } \) approximately. (two decimal places).

Prove that \(\sqrt { \frac { 1-x }{ 1+x } } \) is approximately equal to 1 - x + \(\frac{x^2}{2}\) when x is very small.