# Which term of the progression 201, 192, 183,… is the first negative term?

A progression is basically a list of terms (usually numbers) that follow a particular logical and predictable pattern. There is a certain relation between the two terms in each type of Progression. The predictable nature of Progression helps in forming a generalized formula for that Progression, Formulae include finding the nth term of the series, finding the sum of the series, etc. There are three main types of progressions known,

**Types of Progression**

In Mathematics, the progression of numbers can be classified into three specific types mainly:

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- Arithmetic Progression
- Geometric Progression
- Harmonic Progression

Let’s learn in detail about the arithmetic progression,

**Arithmetic Progression**

Arithmetic Progression is basically a sequence of numbers which exist in such a way that the difference between any two consecutive numbers is a constant value or quantity, that difference is denoted as “d”. The first term in A.P. is denoted as “a” and the last term (for finite series) as “n”. For instance, consider the sequence of even natural numbers 2, 4, 6, 8, 10,…….If we consider the difference between any two numbers (8- 6) is 2. Some of the other few examples of Arithmetic Progression are Sequence of odd natural numbers, Sequence of natural numbers.

**Generalized representation of Arithmetic Progression**

The first term is represented as “a” and the common difference is represented as “d”, therefore, the next term should is a+d, and the next term to that should be a+d+d, based on this, a generalized way of representing the A.P. can be formed. The Arithmetic Progression can be expressed as,

a, a+d, a+2d, a+3d, a+4d, ………. a+ (n-1)d

In the above expression, “a” represents the first term of the progression, “d” represents the common difference

The last term “an” of the progression is represented as,

a

_{n}=a+(n-1)d

### Which term of the progression 201, 192, 183,… is the first negative term?

**Solution:**

From the above equation, it is known that, a

_{1}=201, a_{2}= 192, a_{3}= 183, …..Common difference = a

_{2}– a_{1}= 192 – 201 = -9The task is to find the first negative term i.e. a

_{n }< 0a

_{n}= a + (n-1) da + (n-1)d < 0

201 + (n-1)-9 < 0

201 -9n + 9 < 0

210 -9n < 0

9n > 210

n > 210/9

n > 23.3So, n can be considered as 24 (Approx)

So, 24th term will be the first negative term i.e. 201 + (24-1)-9 = 201 – 207 = -6

**Similar Questions**

**Question 1: Which term of the progression 20, 17, 14,… is the first negative term?**

**Solution:**

From the above equation, it is known that, a

_{1}=20, a_{2}= 17, a_{3}= 14, …..Common difference = a

_{2}– a_{1}= 17 – 20 = -3The task is to find the first negative term i.e. a

_{n}< 0a

_{n}= a + (n-1) d

a + (n-1)d < 0

20 + (n-1)-3 < 0

20 -3n + 3 < 0

23 -3n < 0

3n > 23

n > 23/3

n > 7.6So, n can be considered as 8 (Approx)

So, 8th term will be the first negative term i.e. 20 + (8-1) -3 = 20 -21 = -1.

**Question 2:** **Which term of the progression 121, 117, 113,… is the first negative term?**

**Solution:**

From the above equation, it is known that, a

_{1}=121, a_{2}= 117, a_{3}= 113, …..Common difference = a

_{2}– a_{1}= 117 – 121 = -4The task is to find the first negative term i.e. an < 0

a

_{n}= a + (n-1) d

a + (n-1)d < 0

121 + (n-1)-4 < 0

121 -4n + 4 < 0

125 -4n < 0

4n > 125

n > 125/4

n > 31.25So, n can be considered as 32 (Approx)

So, 32nd term will be the first negative term i.e. 121 + (32-1) -4 = 121-124 = -3.

**Note: **Here we are considering the first term as** a _{1 }**not as

**a**

_{0.}