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-where-1-n-

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

I f p, x_{1}, x_{2}, \dots x_{i} a n d q, y_{1}, y_{2}, \dots 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} + \dots + x_{n}}{n}

α = \frac{(p + a) + (p + 2 a) + \dots + (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 \dots

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 \dots

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 \dots l e t m e c h e c k i n g o i i t

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