Question

answer this question friends ​

Answer 1

Answer:

$1) \: \frac{n(n + 1)}{2}$

EXPLAINATION

FORMULA

$lim_{x - > a}( \frac{ {x}^{n} - {a}^{n} }{x - a} ) = n {a}^{n - 1}$

CALCULATION

$lim_{x - > 1}( \frac{x + {x}^{2} + {x}^{3} + ....... + {x}^{n} - n}{x - 1} )$

$lim_{x - > 1}( \frac{x - 1+ {x}^{2} - 1 + {x}^{3} - 1 + ....... + {x}^{n} - 1}{x - 1} )$

$lim_{x - > 1}( \frac{x - 1}{x - 1} + \frac{ {x}^{2} - 1 }{x - 1} + \frac{ {x}^{3} - 1}{x - 1} + ..... + \frac{ {x}^{n} - 1}{x - 1} )$

$1 + lim_{x - > 1}( \frac{ {x}^{2} - 1}{x - 1} ) + lim_{x - > 1}( \frac{ {x}^{3} - 1}{x - 1} )... + lim_{x - > 1}( \frac{ {x}^{n} - 1 }{x - 1} )$

$1 + 2. {1}^{1} + 3. {1}^{2} + 4. {1}^{3} + ..n. {1}^{n - 1}$

$1 + 2 + 3 + 4 + ...... + n$

$= \frac{n(n + 1)}{2}$