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Question Number 192444 by Mastermind last updated on 18/May/23

The sum of three numbers in arith-  metic progression is 18 and sum of  square is 206. find the numbers

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{three}\:\mathrm{numbers}\:\mathrm{in}\:\mathrm{arith}- \\ $$$$\mathrm{metic}\:\mathrm{progression}\:\mathrm{is}\:\mathrm{18}\:\mathrm{and}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{square}\:\mathrm{is}\:\mathrm{206}.\:\mathrm{find}\:\mathrm{the}\:\mathrm{numbers} \\ $$

Commented by Tinku Tara last updated on 19/May/23

Automatic tagging may sometime  tag wrong question.  will fix this in future updates

$$\mathrm{Automatic}\:\mathrm{tagging}\:\mathrm{may}\:\mathrm{sometime} \\ $$$$\mathrm{tag}\:\mathrm{wrong}\:\mathrm{question}. \\ $$$$\mathrm{will}\:\mathrm{fix}\:\mathrm{this}\:\mathrm{in}\:\mathrm{future}\:\mathrm{updates} \\ $$

Commented by Mastermind last updated on 18/May/23

the answer you tag is already deleted  from forum

$$\mathrm{the}\:\mathrm{answer}\:\mathrm{you}\:\mathrm{tag}\:\mathrm{is}\:\mathrm{already}\:\mathrm{deleted} \\ $$$$\mathrm{from}\:\mathrm{forum} \\ $$

Commented by Frix last updated on 19/May/23

It can be triggered like this:  q=546213

$$\mathrm{It}\:\mathrm{can}\:\mathrm{be}\:\mathrm{triggered}\:\mathrm{like}\:\mathrm{this}: \\ $$$${q}=\mathrm{546213} \\ $$

Commented by Frix last updated on 19/May/23

...or this:  Quod erat demonstrandum x=567123

$$...\mathrm{or}\:\mathrm{this}: \\ $$$$\mathrm{Quod}\:\mathrm{erat}\:\mathrm{demonstrandum}\:{x}=\mathrm{567123} \\ $$

Answered by deleteduser1 last updated on 18/May/23

a−d+a+a+d=3a=18⇒a=6  (6−d)^2 +6^2 +(6+d)^2 =108+2d^2 =206⇒d=+_− 7  (−1,6,13);(13,6,−1)

$${a}−{d}+{a}+{a}+{d}=\mathrm{3}{a}=\mathrm{18}\Rightarrow{a}=\mathrm{6} \\ $$$$\left(\mathrm{6}−{d}\right)^{\mathrm{2}} +\mathrm{6}^{\mathrm{2}} +\left(\mathrm{6}+{d}\right)^{\mathrm{2}} =\mathrm{108}+\mathrm{2}{d}^{\mathrm{2}} =\mathrm{206}\Rightarrow{d}=\underset{−} {+}\mathrm{7} \\ $$$$\left(−\mathrm{1},\mathrm{6},\mathrm{13}\right);\left(\mathrm{13},\mathrm{6},−\mathrm{1}\right) \\ $$

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