Check whether –150 is a term of the AP: 11, 8, 5, 2, …..

Question

Check whether $–150$ is a term of the AP: $11, 8, 5, 2, …..$

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Answer

The given list of numbers is $11, 8, 5, 2,.....$
$ a_2-a_1=8-11=-3 $
$ a_3-a_2=5-8=-3$
$a_4-a_3=2-5=-3$
$\text { i.e. } a_{k+1}-a_k \text { is the same every time. }$
So, the given list of numbers forms an $AP$ with first term $a = 11$ and the common difference $d = -3.$
Let $-150$ be the nth term of the given $AP$
Then, $a_n= -150$
$ \Rightarrow a + (n - 1) d = -150 $
$ \Rightarrow 11+ (n - 1)(-3) = -150$
$ \Rightarrow (-3) (n - 1) = -150 - 11 $
$ \Rightarrow (-3) (n - 1) = -161$
$ \Rightarrow 3(n - 1) = 161$
$ \Rightarrow n - 1 = \frac{{161}}{3}$
$ \Rightarrow n = \frac{{161}}{3} + 1$
$ \Rightarrow n = \frac{{164}}{3}$
But n should be a positive integer. So, $-150$ is not a term of $11, 8, 5, 2,....$

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