Justify whether it is true to say that $ -1,-\frac{3}{2},-2, \frac{5}{2}, \ldots $ forms an AP as $ a_{2}-a_{1}=a_{3}-a_{2} . $
Given:
Given sequence is \( -1,-\frac{3}{2},-2, \frac{5}{2}, \ldots \)
To do:
We have to check whether it is true to say that \( -1,-\frac{3}{2},-2, \frac{5}{2}, \ldots \) forms an AP as \( a_{2}-a_{1}=a_{3}-a_{2} . \)
Solution:
In the given sequence,
$a_1=-1, a_2=-\frac{3}{2}, a_3=-2, a_4=\frac{5}{2}$
$a_2-a_1=-\frac{3}{2}-(-1)=-\frac{3}{2}+1=\frac{-3+2}{2}=\frac{-1}{2}$
$a_3-a_2=-2-(-\frac{3}{2})=--2+\frac{3}{2}=\frac{-4+3}{2}=\frac{-1}{2}$
$a_4-a_3=\frac{5}{2}-(-2)=\frac{5}{2}+2=\frac{5+4}{2}=\frac{9}{2}$
Here,
$a_2 - a_1 = a_3 - a_2$ but $a_4 - a_3≠a_3 - a_2$
Therefore, the given sequence is not an AP.
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