The product of the four terms of an increasing arithmetic progression is a, their sum is b, and the sum of their reciprocal is c. Suppose that a,b,c form a geometric progression whose product is 8000, find the sum of the first and fourth term.


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Question Number 110434 by Aina Samuel Temidayo last updated on 29/Aug/20

$The product of the four terms of an$ $increasing arithmetic progression is a,$ $their sum is b, and the sum of their$ $reciprocal is c . Suppose that a, b, c form$ $a geometric progression whose$ $product is 8000, find the sum of the$ $first and fourth term .$
Commented by Rasheed.Sindhi last updated on 29/Aug/20

$‘ ‘ M o r e t h a n {s u f f i c i e n t}^{″} d a t a i s$ $g i v e n i n t h e q u e s t i o n .$
	Answered by Rasheed.Sindhi last updated on 29/Aug/20

	$L e t t h e t e r m s a r e :$ $u - 3 d, u - d, u + d, u + 3 d; d > 0$ $H a v e t o d e t e r m i n e :$ $W e^{'} v e t o d e t e r m i n e s u m o f f i r s t$ $& f o u r t h t e r m .$ $T h e r e q u i r e d s u m$ $= (u - 3 d) + (u + 3 d) = 2 u$ $S o {w e}^{'} v e t o d e t e r m i n e u$ $G i v e n :$ $Extra \left or missing \right$ $(u^{2} - d^{2}) (u^{2} - 9 d^{2}) = a (i)$ $Extra \left or missing \right$ $4 u = b \Rightarrow u = b / 4 (i i)$ $^{∙ 3} \frac{1}{u - 3 d} + \frac{1}{u - d} + \frac{1}{u + d} + \frac{1}{u + 3 d} = c$ $(A s t h e g i v e n d a t a i n t h i s q u e s t i o n$ $i s ‘ ‘ m o r e t h a n {s u f f i c i e n t}^{″} . S o t h i s$ $p i e c e o f i n f o r m a t i o n i s n e g l e c t e d h e r e .)$ $^{∙ 4} a b c = 8000 \Rightarrow a c = 8000 / b (i i i)$ $^{∙ 5} b^{2} = a c \Rightarrow b^{2} = 8000 / b \Rightarrow b^{3} = 8000$ $b = 20$ $a c = 8000 / b = 8000 / 20 = 400$ $(i i) \Rightarrow u = b / 4 = 20 / 4 = 5$ $S u m o f f i r s t & f o u r t h t e r m s$ $= (u - 3 d) + (u + 3 d) = 2 u = 2 (5)$ $= 10$
	Commented by Aina Samuel Temidayo last updated on 29/Aug/20

	$Thanks, but how did you know more$ $than sufficient data is given in the$ $question ?$
	Commented by Rasheed.Sindhi last updated on 29/Aug/20

	$W h e n w e c a n s o l v e a p r o b l e m$ $w i t h o u t u s i n g a g i v e n$ $i n f o r m a t i o n p i e c e i t m e a n s t h a t$ $t h i s p i e c e o f i n f o r m a t i o n i s$ $n o t n e c e s s a r y i - e {i t}^{'} s m o r e t h a n$ $s u f f i c i e n t .$
	Commented by Aina Samuel Temidayo last updated on 30/Aug/20

	$Thanks .$
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