Which term of the progression 201, 192, 183,β¦ 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:
- Arithmetic Progression
- Geometric Progression
- Harmonic Progression
Letβs 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 βdβ. The first term in A.P. is denoted as βaβ and the last term (for finite series) as βnβ. For instance, consider the sequence of even natural numbers 2, 4, 6, 8, 10,β¦β¦.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 βaβ and the common difference is represented as βdβ, 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, β¦β¦β¦. a+ (n-1)d
In the above expression, βaβ represents the first term of the progression, βdβ represents the common difference
The last term βanβ of the progression is represented as,
an=a+(n-1)d
Which term of the progression 201, 192, 183,β¦ is the first negative term?
Solution:
From the above equation, it is known that, a1 =201, a2 = 192, a3 = 183, β¦..
Common difference = a2β a1 = 192 β 201 = -9
The task is to find the first negative term i.e. an < 0
an = a + (n-1) d
a + (n-1)d < 0
201 + (n-1)-9 < 0
201 -9n + 9 < 0
210 -9n < 0
9n > 210
n > 210/9
n > 23.3
So, n can be considered as 24 (Approx)So, 24th term will be the first negative term i.e. 201 + (24-1)-9 = 201 β 207 = -6
Similar Questions
Question 1: Which term of the progression 20, 17, 14,β¦ is the first negative term?
Solution:
From the above equation, it is known that, a1 =20, a2 = 17, a3 = 14, β¦..
Common difference = a2β a1 = 17 β 20 = -3
The task is to find the first negative term i.e. an < 0
an = 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,β¦ is the first negative term?
Solution:
From the above equation, it is known that, a1 =121, a2 = 117, a3 = 113, β¦..
Common difference = a2β a1 = 117 β 121 = -4
The task is to find the first negative term i.e. an < 0
an = 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 a1 not as a0.