Question

Find ${1^{st}}$ negative term of \[231, 228, 225, \ldots \]

Answer 1

Hint:
Here, we will verify if the given sequence is an arithmetic progression or not by subtracting each term and its preceding term. If the common difference turns out to be the same, thus, it is an AP. Then, we will use the general formula of an AP and find the terms which is 0. This term comes just before the first negative term. Hence, next term will give us the required ${1^{st}}$ negative term of \[231,228,225, \ldots \]

Formula Used:
General term of an AP is: $a + \left( {n - 1} \right)d$, where $a$ is the first term, $d$ is the common difference and $n$ is the number of the term.

Complete step by step solution:
Given sequence is: \[231,228,225, \ldots \]
First, we will find whether this given sequence is an Arithmetic Progression or not.
We will subtract each term and its preceding term, therefore, we get
$228 - 231 = - 3$
$225 - 228 = - 3$
We can see that the difference between each term and its preceding term is same, thus this sequence forms an Arithmetic Progression (AP)
The first term of this AP, $a = 231$
The common difference, $d = - 3$
Now, in order to find the ${1^{st}}$ negative term, first, we will find the term which gives us zero.
Hence, let the ${n^{th}}$ term of this AP be 0.
Hence, since, the ${n^{th}}$ term of this AP is 0. Thus, equating it with the general term $a + \left( {n - 1} \right)d$, we get,
$a + \left( {n - 1} \right)d = 0$
Now, substituting the known values of first term and common difference in the above equation, we get
$231 + \left( {n - 1} \right)\left( { - 3} \right) = 0$
Multiplying the terms, we get
$ \Rightarrow 231 - 3n + 3 = 0$
Adding the like terms, we get
$ \Rightarrow 3n = 234$
Dividing both sides by 3, we get
$ \Rightarrow n = 78$
Therefore, ${78^{th}}$ term of this AP is 0.
Thus, since, the terms of this AP are decreasing, thus, the ${79^{th}}$ term of this AP will definitely be a negative number.
Using the general form, ${79^{th}}$ term of an AP can be written as $a + \left( {79 - 1} \right)d$.
Now, substituting the given values, we get,
${79^{th}}$term$ = 231 + 78\left( { - 3} \right)$
Multiplying the terms, we get
${79^{th}}$term $ = 231 - 234 = - 3$

Therefore, the required ${1^{st}}$ negative term of \[231, 228, 225, \ldots \] is $ - 3$
Hence, $ - 3$ is the required answer.

Note:
An Arithmetic Progression is a sequence of numbers such that the difference between any term and its preceding term is constant. This difference is known as the common difference of an Arithmetic Progression (AP). A real-life example of AP is when we add a fixed amount in our money bank every week. Similarly, when we ride a taxi, we pay an amount for the initial kilometer and pay a fixed amount for all the further kilometers, this also turns out to be an AP.