Josephina thinks that she has discovered a new pattern, shown at right.

1. Check her equations by multiplying. Does her pattern work?

   Hint (a):

   To check a division problem using multiplication, multiply the divisor by the quotient. In the second case, multiply $(x - 1)(x + 1)$. Does it equal $x^2 - 1?$ How will you check the third and fourth cases?

2. Based on her pattern, what does $\frac { x ^ { 5 } - 1 } { x - 1 }$ equal? Is it true? Justify your answer.

   Hint (b):

   Look for a pattern in the quotients. Look at the number of terms and the exponent each term.

3. Make a conjecture about how to represent the result for $\frac { x ^ { n } - 1 } { x - 1 }$. You will need “…” in the middle of your expression.

   Answer (c):

   $x^n + x^{n-1} + x^{n-2} +...+ x + 1$

$\frac{x-1}{x-1}=1$
$\frac{x^2-1}{x-1}=x+1$
$\frac{x^3-1}{x-1}=x^2+x\ +1$
$\frac{x^4-1}{x-1}=x^3+x^2+x+1$