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Question Number 174533 by mnjuly1970 last updated on 03/Aug/22

Answered by behi834171 last updated on 03/Aug/22

$${\mathit{x}}^2=2\mathit{x}+1\Rightarrow {\mathit{x}}^2-2\mathit{x}-1=0\Rightarrow \{\begin{array}{l}\mathit{\alpha }+\mathit{\beta }=2\\ \mathit{\alpha }.\mathit{\beta }=-1\end{array}$$$${\mathit{x}}^3=2{\mathit{x}}^2+\mathit{x}=5\mathit{x}+2$$$${\mathit{x}}^4=5{\mathit{x}}^2+2\mathit{x}=10\mathit{x}+5+2\mathit{x}=12\mathit{x}+5$$$${\mathit{x}}^5=12{\mathit{x}}^2+5\mathit{x}=24\mathit{x}+12+5\mathit{x}=29\mathit{x}+12$$$${\mathit{x}}^6=29{\mathit{x}}^2+12\mathit{x}=58\mathit{x}+29+12\mathit{x}=70\mathit{x}+29$$$${\mathit{x}}^7=70{\mathit{x}}^2+29\mathit{x}=140\mathit{x}+70+29\mathit{x}=169\mathit{x}+70$$$$\Rightarrow A=(2\mathit{\alpha }+1).(169\mathit{\beta }+70)+15(2\mathit{\alpha }+1)+\mathit{\beta }+3=$$$$=338\alpha \beta +140\alpha +169\beta +70+30\alpha +15+\beta +3=$$$$=338\alpha \beta +170(\alpha +\beta )+88=$$$$=-338\times 1+170\times 2+88=90.◼$$

Commented by Rasheed.Sindhi last updated on 03/Aug/22

$$◼\mathrm{Nice}\mathrm{sir}!$$

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