Question

Which term of the A.P $150$ , $147$ , $144$ ,…… is its first negative term?

Answer 1

In this problem we need to find which term of the given A.P is the first negative term. So we will first consider the given A.P and compare it with ${{a}_{0}}$ , ${{a}_{1}}$ , ${{a}_{2}}$ ,….. and calculate the value of common difference $d$ which is given by ${{a}_{1}}-{{a}_{0}}$ or ${{a}_{2}}-{{a}_{1}}$ . Now we will first check whether the given A.P has a zero value or not by using the formula ${{a}_{n}}={{a}_{0}}+\left( n-1 \right)d$ . In this formula we will substitute ${{a}_{n}}=0$ and calculate the value of $n$ by using basic mathematical operations. If we get the $n$value as a natural number, then the upcoming term in the A.P is the first negative term.

Complete step by step answer:
Given Arithmetic Progression is $150$ , $147$ , $144$ ,……
Comparing the above Arithmetic Progression with ${{a}_{0}}$ , ${{a}_{1}}$ , ${{a}_{2}}$ ,….., then we will have
${{a}_{0}}=150$ , ${{a}_{1}}=147$ , ${{a}_{2}}=144$ .
Now the common difference $d$ of the progression is equal to the difference between the two successive terms, then we will get
$\begin{aligned} & d={{a}_{1}}-{{a}_{0}}\text{ or }{{a}_{2}}-{{a}_{1}} \\\ & \Rightarrow d=147-150\,\text{ or }144-147 \\\ & \Rightarrow d=-3 \\\ \end{aligned}$
We are going to check whether the given Arithmetic Progression has $0$ or not by substituting ${{a}_{n}}=0$ in the formula ${{a}_{n}}={{a}_{0}}+\left( n-1 \right)d$, then we will get
$0=150+\left( n-1 \right)\left( -3 \right)$
Simplifying the above equation by applying basic mathematical operations, then we will have
$\begin{aligned} & 0=150-\left( n-1 \right)3 \\\ & \Rightarrow 3\left( n-1 \right)=150 \\\ & \Rightarrow n-1=\dfrac{150}{3} \\\ & \Rightarrow n=50+1 \\\ & \Rightarrow n=51 \\\ \end{aligned}$
From the above value we can say that ${{51}^{st}}$ term of the given Arithmetic Progression is zero. So the next term which is ${{52}^{nd}}\left( 51+1 \right)$ terms is definitely the first negative term of the given Arithmetic Progression.

Note: In this problem when we are checking whether the given progression has zero or not. We have the value of $n$ as $51$ which is a natural number. There is no guarantee that each progression has zero or for each progression we can get the value of $n$as a natural number. In some cases we may get a fraction also. In that case we will consider $n$value to the nearest natural number and check whether it is a negative number or not.