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Easy

Question

The sum of the series $1 times n plus 1 plus 2 times open parentheses n minus 1 close parentheses plus 3 times open parentheses n minus 2 close parentheses horizontal ellipsis plus n times 1$ is

$\frac{n (n + 1) (n + 2)}{6}$
$\frac{n (n + 1) (n + 2)}{3}$
$\frac{n (n + 1) (3 n + 2)}{6}$
$\frac{n (n + 1) (2 n + 2)}{3}$

The correct answer is: $fraction numerator n open parentheses n plus 1 close parentheses left parenthesis n plus 2 right parenthesis over denominator 6 end fraction$

$1 bullet n plus 2 open parentheses n minus 1 close parentheses plus 3 open parentheses n minus 2 close parentheses plus... plus n bullet 1$
$equals not stretchy sum from r equals 1 to n of open parentheses n plus 1 close parentheses r minus not stretchy sum from r equals 1 to n of r to the power of 2 end exponent$
$equals open parentheses n plus 1 close parentheses not stretchy sum n minus not stretchy sum n to the power of 2 end exponent$
$equals fraction numerator open parentheses n plus 1 close parentheses n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction minus fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction$
$equals fraction numerator n open parentheses n plus 1 close parentheses over denominator 6 end fraction open curly brackets 3 n plus 3 minus 2 n minus 1 close curly brackets$
$equals fraction numerator n open parentheses n plus 1 close parentheses left parenthesis n plus 2 right parenthesis over denominator 6 end fraction$

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parallel

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The line $L subscript 1 end subscript colon y minus x equals 0 text end text$ and $text end text L subscript 2 end subscript colon 2 x plus y equals 0$ intersect the line $L subscript 3 end subscript colon y plus 2 equals 0$ at P and Q respectively. The bisector of the acute angle between L₁ and L₂ intersects L₃at R
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Statement - 2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.

parallel

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