Question

Let x= -1

Find x + x^2 + x^3 + ......+ x^2010 + x^2011
(The dots mean the numbers between 3 and 2010.)

Answer 1

`\(x + x^2 + x^3 + \ldots + x^(2010) + x^(2011) = -1\).`

Explanation:

The expression
`\(x + x^2 + x^3 + \ldots + x^(2010) + x^(2011)\)` represents a finite geometric series.

The sum of a finite geometric series with the first term a, common ratio r, and n terms is given by the formula:

`\[S_n = (a(r^n - 1))/(r - 1)\]`

In this case,
`\(a = x\), \(r = x\), and \(n = 2011\)`. Plug these values into the formula:

`\[S_(2011) = (x(x^(2011) - 1))/(x - 1)\]`

Now, substitute \(x = -1\):

`\[S_(2011) = ((-1)((-1)^(2011) - 1))/((-1) - 1)\]`

Since
`\((-1)^(2011) = -1\),` the expression becomes:

`\[S_(2011) = (-1(-1 - 1))/(-2)\]`

Simplify the expression:

`\[S_(2011) = (-1 \cdot (-2))/(-2)\]`

`\[S_(2011) = (2)/(-2)\]`

`\[S_(2011) = -1\]`

So,
`\(x + x^2 + x^3 + \ldots + x^(2010) + x^(2011) = -1\).`