Question

Check whether $-150$ is a term of A.P. $11,8,5,2......$

Answer 1

We are given the arithmetic progression sequence, we will apply the conventional formula for ${{n}^{th}}$ term, i.e ${{a}_{n}}=a+\left( n-1 \right)d$ and then we will find the value of $d$ first. After that we will again put the given number that is $-150$ into the formula for ${{n}^{th}}$ term and then find the value of $n$ . If the value of $n$ is positive $\left( >0 \right)$ then the given number is a term of the given sequence.

Complete step by step answer:
Since it is given that the sequence $11,8,5,2......$ is in arithmetic progression, therefore it will be in the following form:
$a,a+d,a+2d,.......,a+\left( n-1 \right)d,.....$
Where, ${{a}_{n}}=a+\left( n-1 \right)d$ is the ${{n}^{th}}$ term and $a$ is the first term in the sequence and $d$ is the common difference between terms.
We will first find out the value of $d$, for this we will start by taking $n=4$ , and now we will use the formula for ${{n}^{th}}$ term and since the ${{4}^{th}}$ term in the sequence is $2$, we will take ${{a}_{n}}=2$ and of course, we can see that the first term of the sequence is $a=11$. Now putting all these in the formula we will get the following:
$\begin{aligned} & {{a}_{n}}=a+\left( n-1 \right)d\Rightarrow 2=11+\left( 4-1 \right)d\Rightarrow -9=3d \\\ & \Rightarrow d=-3 \\\ \end{aligned}$
Now since we have our value of $d$ , to check that whether $-150$ is a term of A.P. $11,8,5,2......$ or not, we will take ${{a}_{n}}=-150$ and find the value of $n$:
$\begin{aligned} & {{a}_{n}}=a+\left( n-1 \right)d \\\ & \Rightarrow -150=11+\left( n-1 \right)\left( -3 \right)\Rightarrow -161=\left( n-1 \right)\left( -3 \right)\Rightarrow \dfrac{161}{3}=n-1 \\\ & \Rightarrow \dfrac{164}{3}=n \\\ \end{aligned}$

Since, the value of $n$ cannot be in fraction that is why $-150$ is not a term of given sequence.

Note: You can also find the value of $d$ by the following method that is: $d={{a}_{n}}-{{a}_{n-1}}$ . Just take any two terms and then subtract them. Since, the calculation is not that tedious, try and explain your solution and each step clearly for example at last after finding out the value of $n$ in fraction, explain why it is not valid.