If p , x_1 ,x_2 ,...x_i and q,y_1 ,y_2 ,...y_i form two infinite arithmetic sequences with common difference a and b respectively , then find the locus of the point ( α , β ) where α = (1/n) Σ_(i=1) ^n x_i and β= (1/n) Σ_(i=1) ^n y


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Question Number 61274 by alphaprime last updated on 31/May/19

$I f p, x_{1}, x_{2}, . . . x_{i} a n d q, y_{1}, y_{2}, . . . y_{i} f o r m t w o$ $i n f i n i t e a r i t h m e t i c s e q u e n c e s w i t h c o m m o n$ $d i f f e r e n c e a a n d b r e s p e c t i v e l y,$ $t h e n f i n d t h e l o c u s o f t h e p o i n t (α, β)$ $w h e r e α = \frac{1}{n} \sum_{i = 1}^{n} x_{i} a n d β = \frac{1}{n} \sum_{i = 1}^{n} y_{i .}$
	Answered by tanmay last updated on 31/May/19

	$α = \frac{x_{1} + x_{2} + x_{3} + . . . + x_{n}}{n}$ $α = \frac{(p + a) + (p + 2 a) + . . . + (p + n a)}{n}$ $α = \frac{n p + a [\frac{n (n + 1)}{2}]}{n}$ $α = p + \frac{a (n + 1)}{2}$ $β = q + \frac{b (n + 1)}{2}$ $\frac{α - p}{\frac{a}{2}} = \frac{β - q}{\frac{b}{2}}$ $s o l o c u s . . .$ $b (x - p) = a (y - q)$ $b x - a y + a q - b p = 0$ $e a n o f s t r a i g h t l i n e \to l o c u s s t l i n e . . .$
	Commented by bhanukumarb2@gmail.com last updated on 31/May/19

	$s i r c o n g r a t e s f o r g o i i t a p p r o v a l p l e a s d$ $s h a r e u r a p p r o c h t h e i r a l s o$
	Commented by bhanukumarb2@gmail.com last updated on 31/May/19

	$t a n m a y s i r$
	Commented by tanmay last updated on 31/May/19

	$t h a n k y o u s i r . . . l e t m e c h e c k i n g o i i t$
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