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Question Number 34870 by Tinkutara last updated on 12/May/18

Commented by Tinkutara last updated on 15/May/18

please help

Commented by Tinkutara last updated on 16/May/18

Anyone please see it.

Commented by Tinkutara last updated on 16/May/18

Commented by Tinkutara last updated on 16/May/18

Actually this was the whole comprehension.

Answered by Rasheed.Sindhi last updated on 20/May/18

$$\mathrm{Pattern}\mathrm{approach}$$$$\text{You can't use 'macro parameter character \#' in math mode}$$$$\mathrm{determine}{}^{\prime }\mathrm{number}\mathrm{of}{\mathrm{triangles}}^{\prime }\mathrm{for}$$$$\mathrm{various}\mathrm{values}\mathrm{of}\mathrm{c}:(\mathrm{In}\mathrm{the}\mathrm{following}\mathrm{N}$$$$\mathrm{denotes}\mathrm{number}\mathrm{of}\mathrm{triangle}.)$$$$\left(\begin{array}{cc}\mathrm{c}& \mathrm{N}\\ 1& 3\\ 2& 12\\ 3& 27\\ 4& 48\\ 5& 75\\ ⋮& ⋮\end{array}\right)$$$$\mathrm{Number}\mathrm{of}\mathrm{triangles}\left(\mathrm{N}\right)\mathrm{make}\mathrm{following}$$$$\mathrm{sequence}:$$$$3,12,27,48,75,108,....$$$$\mathrm{We}\mathrm{can}\mathrm{easily}\mathrm{determine}\mathrm{general}\mathrm{term}:$$$$\left[\begin{array}{cccccccc}\mathrm{c}& 1& 2& 3& 4& 5& 6& ..\\ \mathrm{N}& 3& 12& 27& 48& 75& 108& ..\\ & 3.1& 3.4& 3.9& 3.16& 3.25& 3.36& ..\\ & 3.1^2& 3.2^2& 3.3^2& 3.4^2& 3.5^2& 3.6^2& ..\end{array}\right]$$$$\mathrm{Hence}\mathrm{general}\mathrm{term}\mathrm{is}{3\mathrm{c}}^2$$$$\mathrm{Number}\mathrm{of}\mathrm{triangles}:{3\mathrm{c}}^2$$

Answered by Tinkutara last updated on 22/May/18

Commented by Tinkutara last updated on 22/May/18

This was the solution. Can you please elaborate it? I can't understand it.

Answered by Rasheed.Sindhi last updated on 24/May/18

$$\mathrm{A}\mathrm{try}\mathrm{to}\mathrm{elaborate}\mathrm{the}\mathrm{answer}\mathrm{given}$$$$\mathrm{with}\mathrm{question}\mathrm{in}\mathrm{the}\mathrm{book}.$$$$\left(\mathrm{See}\mathrm{ans}\mathrm{by}\mathrm{Mr}\mathrm{Tinkutara}\mathrm{here}\right)$$$$(a,a,b):\mathrm{A}\mathrm{triangle},\left(\mathrm{i}\right)\mathrm{with}\mathrm{integer}\mathrm{sides}$$$$\mathrm{and}\left(\mathrm{ii}\right)\mathrm{at}\mathrm{least}\mathrm{two}\mathrm{sides}\mathrm{are}\mathrm{equal}\mathrm{with}$$$$\mathrm{the}\mathrm{possibility}a=b.$$$$\mathrm{By}\mathrm{triangle}\mathrm{inequality},$$$$a+a>b\Rightarrow 2a-1⩾b$$$$\mathrm{Also}b⩾1$$$$\mathrm{So},$$$$2a-1⩾b⩾1...............\mathrm{A}$$$$1⩽a,b⩽2\mathrm{c}\left(\mathrm{Given}\right).......\mathrm{B}$$$$\mathrm{Case}-1:1⩽a⩽\mathrm{c}$$$$1⩽b⩽2c\wedge b⩽2a-1\Rightarrow 1⩽b⩽2a-1\ast$$$$\mathrm{i}-\mathrm{e}\mathrm{there}\mathrm{are}2a-1\mathrm{choices}\mathrm{for}b\mathrm{when}$$$$a\mathrm{is}\mathrm{fixed}.$$$$\mathrm{For}a=c\to 2c-1choicesforb$$$$\mathrm{For}a=c-1\to 2(c-1)-1=2c-3choicesforb$$$$\mathrm{For}a=c-2\to 2(c-2)-1=2c-5choicesforb$$$$......$$$$.....$$$$\mathrm{For}a=2\to 2\left(2\right)-1=3choicesforb$$$$\mathrm{For}a=1\to 2\left(1\right)-1=1choicesforb$$$$\mathrm{Total}\mathrm{choices}\mathrm{for}\mathrm{this}\mathrm{case}$$$$\underset{------------cterms------}{(2\mathrm{c}-1)+(2\mathrm{c}-3)+(2\mathrm{c}+5)...+3+1}=c^2$$$$\mathrm{Notice}\mathrm{that}\mathrm{above}\mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{c}\mathrm{consecutive}$$$$\mathrm{odd}\mathrm{numbers}.\left(\mathrm{Treate}\mathrm{it}\mathrm{as}\mathrm{AP}\right)$$$$\mathrm{Case}-2:\mathrm{c}+1⩽a⩽2\mathrm{c}\left(\mathrm{c}\mathrm{values}\right)$$$$b⩽2\mathrm{c}\wedge 1⩽b⩽2a-1\Rightarrow 1⩽b⩽2c\ast \ast$$$$\mathrm{i}-\mathrm{e}b\mathrm{takes}2\mathrm{c}\mathrm{values}.$$$$\because a\mathrm{takes}c\mathrm{values}\mathrm{from}\mathrm{c}+1\mathrm{to}2\mathrm{c}$$$$\mathrm{and}\mathrm{for}\mathrm{each}\mathrm{of}\mathrm{these}\mathrm{values}$$$$b\mathrm{takes}2c\mathrm{vales}$$$$\therefore (a,a,b)\mathrm{takes}c.2c=2c^2\mathrm{values}$$$$\mathrm{Case}-1\mathrm{\&}\mathrm{Case}-2:$$$$c^2+2c^2=3c^2$$$$Numberof(a,a,b)=3c^2$$

Commented by Rasheed.Sindhi last updated on 24/May/18

$$\ast \mathrm{For}1⩽a⩽c$$$$b⩽2c\wedge b⩽2a-1$$$$\Rightarrow b⩽2a-1$$$$[b⩽2c\wedge b⩽2a-1\Rightarrow b⩽\min (2c,2a-1)$$$$1⩽a⩽c\Rightarrow 2⩽2a-1⩽2c-1$$$$\Rightarrow 2a-1⩽2c$$$$\mathrm{So}2a-1\mathrm{is}\mathrm{always}\mathrm{less}\mathrm{than}2c]$$$$\ast \ast \mathrm{For}c+1⩽a⩽2c$$$$b⩽2c\wedge b⩽2a-1$$$$\Rightarrow b⩽2c$$$$[b⩽2c\wedge b⩽2a-1\Rightarrow b⩽\min (2c,2a-1)$$$$c+1⩽a⩽2c\Rightarrow 2c⩽2a-1⩽4c-1$$$$\mathrm{So}2c\mathrm{always}\mathrm{less}\mathrm{than}2a-1$$$$$$

Commented by Tinkutara last updated on 24/May/18

Thank you very much Sir! I got the answer. ��������

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