Prove that the sum of first n even numbers is n(n+1).

Prove that 2+4+6+8+.....2=n(n+1).

Prove that 2+6+18+.....2.3^n-1 = (3ⁿ - 1) for all \(n\in N\).

Prove that 1+2+2²+...+2ⁿ = 2ⁿ⁺¹ 1 for all natural numbers n.

Using principle of mathematical induction, prove that \(\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) .....\left( 1+\frac { 1 }{ n } \right) =n+1\)

Using principle of mathematical induction, prove that
\(\frac { 1 }{ 2.5 } +\frac { 1 }{ 5.8 } +\frac { 1 }{ 8.11 } +....+\frac { 1 }{ (3n-1)(3n+2) } =\frac { n }{ 6n+4 } \)

Prove that 2²ⁿ -1 is divisible by 3, for all natural numbers n.

Prove that \(\sum _{ t=1 }^{ n-1 }{ t(t+1) } =\frac { n(n-1)(n+1) }{ 3 } \) , for all natural numbers \(n\ge 2\)

Simplify the following
i^-39

Simplify the following
i⁹+i¹⁹

Prove by mathematical induction that \({ 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 }+...+{ n }^{ 3 }=\left[ \frac { n(n+1) }{ 2 } \right] ^{ 2 }\), for all \(n\varepsilon N\)

Prove by the principle of mathematical induction that \(1+3+3^{ 2 }+....+3^{ n-1 }=\frac { 3^{ n }-1 }{ 2 } \) , for all \(n \in N\).
 

Prove the rule of exponents (ab)ⁿ=aⁿ bⁿ  by using principle of mathematical induction fo every natural number.

Prove by the principle of mathematical induction that log xⁿ =n log x, for \(n \in N\)

Prove by the principle of mathematical induction that \(1\times 1!+2\times 2!+3\times 3!+....+n\times n!=(n+1)!-1\)for all natural numbers n.

Find the value of \(\sqrt { -25 } +3\sqrt { -4 } +2\sqrt { -9 } \)

Write the real and imaginary parts of the complex number \(\sqrt { 37 } +\sqrt { -19 }\) 
Write the number \(\sqrt { 37 } +\sqrt { -19 }\) in the form z = a + ib and compare, we get Re(z) = a and Im(z) = b

Express the following in the form of a + ib.
\(\left( \frac { 1 }{ 2 } +\frac { 5 }{ 2 } i \right) -\frac { 3 }{ 2 } i+\left( -\frac { 5 }{ 2 } -i \right) \)

Find the real values of x and y, if (x⁴ + 2xi) - (3x² + iy) = (3 - 5i) + (1 + 2iy).
(i) Firstly, separate real and imaginary parts of both sides.
(ii) Second, equate the real and imaginary parts of both sides and get equations in terms of x and y.
(iii) Further, solve these equations to get the values of x and y.

Express the following in the form a + ib.
\(\left( -i \right) \left( 3i \right) { \left( -\frac { 1 }{ 6 } i \right) }^{ 3 }\)

Find the conjugate of \(\frac { (3-2i)(2+3i) }{ (1+2i)(2-i) }\)

Find real values of x and y for which the complex numbers - 3 + ix² y and x² + y + 4i are conjugate of each other.

Convert the complex numbers in polar form  1 - i

If x + iy = \(\frac { a+ib }{ a-ib } \) prove that x² + y² = 1.

If a + ib = \(\frac { ({ x }^{ 2 }+1) }{ 2{ x }^{ 2 }+1 } \) , prove that \({ a }^{ 2 }+{ b }^{ 2 }=\frac { ({ x }^{ 2 }+1)^{ 2 } }{ (2{ x }+1)^{ 2 } } \)