+0  
 
+2
102
1
avatar+481 

If    \({△}_{k}=\begin{vmatrix} 2.3^{k-1} && 3.4^{k-1} && 4.5^{k-1} \\ \alpha && \beta && \gamma \\ 3^n-1 && 4^n-1 && 5^n-1 \end{vmatrix}\\ \)  then the value of \(\sum_{n}^{k=1}{△}_{k} \)  depends

​1) only on \(\alpha, \beta\) and not on \(\gamma\)

2) on none of \(\alpha, \beta\) and \(\gamma\)

3) on all of \(\alpha, \beta \) and \(\gamma\)

4) only on \(\alpha \) not on \(\beta\) and \(\gamma\)

 

Also give possible explanation for the same. \(\)

 May 7, 2021
 #1
avatar+32312 
+4

It looks like 2) is the answer.  

I used the symbolic mathematics part of Mathcad to do this.  I guess that with a lot of tedious manipulation by hand you would be able to show the coefficients of alpha, beta and gamma are all zero.  I didn't see any nice shortcut!

 May 14, 2021

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