If A, B, C, D are consecutive terms in an arithmetic progression, what is the value of \(\frac{ D^2 - A^2 } { C^2 - B^2}\)
Assume B^2 is not equal to C^2.
If A, B, C, D are consecutive terms in an arithmetic progression,
what is the value of \(\dfrac{ D^2 - A^2 } { C^2 - B^2}\)
Assume \(B^2\) is not equal to \(C^2\).
AP:
\(\begin{array}{|rcll|} \hline A &=& a \\ B &=& a+d \\ C &=& a+2d \\ D &=& a+3d \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{ D^2 - A^2 } { C^2 - B^2}} &=& \dfrac{(D-A)(D+A)} { (C-B)(C+B)} \\\\ &=& \dfrac{(a+3d-a)(a+3d+a)} { \Big(a+2d-(a+d)\Big)(a+2d+a+d)} \\\\ &=& \dfrac{3d(2a+3d)} { (a+2d-a-d)(2a+3d)} \\\\ &=& \dfrac{3d(2a+3d)} { d(2a+3d)} \\\\ &=& \mathbf{3} \\ \hline \end{array}\)