leg A_1 ,A_2 ,...A_n and H_1 ,H_2 ,...H_n are n A.M′S and H.M′S respectively between a and b prove that A_r H


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Question Number 58248 by tanmay last updated on 20/Apr/19

$l e g A_{1}, A_{2}, . . . A_{n} a n d H_{1}, H_{2}, . . . H_{n} a r e n A . M^{'} S$ $a n d H . M^{'} S r e s p e c t i v e l y b e t w e e n a a n d b$ $p r o v e t h a t A_{r} H_{n - r + 1} = a b$ $n ⩾ r ⩾ 1$
	Answered by Kunal12588 last updated on 20/Apr/19

	$a, A_{1}, A_{2}, . . . A_{n}, b$ $a r e i n A P$ $w h e r e$ $T_{1} = a,$ $T_{n + 2} = b$ $\Rightarrow T_{1} + (n + 1) d = b$ $\Rightarrow d = \frac{b - a}{n + 1}$ $A_{r} = T_{1} + r d$ $\Rightarrow A_{r} = a + r (\frac{b - a}{n + 1})$ $\Rightarrow A_{r} = \frac{a n + a + r b - r a}{n + 1}$ $\Rightarrow A_{r} = \frac{a (n - r + 1) + r b}{n + 1}$ $a, H_{1}, H_{2}, . . . H_{n}, b$ $a r e i n H P$ $\Rightarrow \frac{1}{a}, \frac{1}{H_{1}}, \frac{1}{H_{2}}, . . ., \frac{1}{H_{n}}, \frac{1}{b} a r e i n A P$ $\Rightarrow \frac{1}{H_{r}} = \frac{\frac{1}{a} (n - r + 1) + r \frac{1}{b}}{n + 1} = \frac{b (n - r + 1) + r a}{a b (n + 1)}$ $\Rightarrow H_{r} = \frac{a b (n + 1)}{b (n - r + 1) + r a}$ $H_{n - r + 1} = \frac{a b (n + 1)}{b (n - (n - r + 1) + 1) + a (n - r + 1)}$ $\Rightarrow H_{n - r + 1} = \frac{a b (n + 1)}{b r + a (n - r + 1)}$ $∴ A_{r} H_{n - r + 1} = \frac{a (n - r + 1) + r b}{n + 1} \times \frac{a b (n + 1)}{b r + a (n - r + 1)}$ $\Rightarrow A_{r} H_{n - r + 1} = a b (1)$ $\Rightarrow A_{r} H_{n - r + 1} = a b$
	Commented by tanmay last updated on 20/Apr/19

	$e x c e l l e n t . . . i h a v e p o s t e d t h e s e q u e s t i o n [f r o m$ $o l d b o o k o f B S T A T q u e s t i o n s . . .$
	Commented by Kunal12588 last updated on 20/Apr/19
	Sir I am in class 12. I got 70.8% in class 11, I know it's not good. What should I do after 12th. I am ok in maths, not so good in physics and very bad in chemistry.
	Commented by tanmay last updated on 20/Apr/19

	$s o f a r i k n o w i n t o d a y s s c e n a r i o p l e n t y o f$ $e n g i n e e r s a n d d o c t o r s a r e t h e r e . . . b u t t h e y a r e$ $n o t s a t i s f i e d w h a t e v e r t h e y g e t i n r e t u r n . . .$ $y o u m a y s e e y o u t u b e v i d e o t h e h i d d e n m e a n i n g$ $o f c o m p a n y p a c k a g e o f f e r t o I I T p a s s o u t . .$ $s o l i s t e n t o y o u r i n n e r o n e v o i c e w h a t h e w a n t . .$ $i a m 49 y e a r o l d g o v t e m p l o y e e m y % a g e$ $w a s n e v e r h i g h b u t i w a s g o o d i n m a t h . . w h a t$ $i a m t o d a y b l e s s i n g o f m a t h . . .$ $s o i t h i n k . . 1) y o u m a y t r y N D A (n a t i o n a l d e f e n c e a c a d e m y)$ $e x a m b a s e d o n 10 + 2 p h y + c h e m + m a t h$ $2) A p p e a r i n B S T A T (B a c h e l o r o f s t a t i c t i c s) e x a m$ $o n M a t h o n l y$ $3) T a r g e t I A S e x a m a n d g e t t r a i n i n g i n D e l h i$ $e v e r y b o d y a r e n o t n u m b e r o n e . . .$ $b u t t h e y s h o u l d j u d g e t h e m s e l f . . w h a t i s f i t$ $f o r t h e m . . .$ $f i n a l l y b u y a b o o k o n c a r e e r c o n s u l t a t i o n o r$ $l i s t e n e x p e r t s o p i n i o n o n y o u t u b e .$
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