Which term of the AP 21, 18, 15 is zero?

Progression is a sequence of numbers that are arranged in a particular pattern. Following the pattern, other numbers in the sequence can also be determined.

Let’s understand this clearly with an example:
Consider the sequence 1,4,9,16,25,……..
Upon carefully observing we can clearly see that each term is nothing but a square of its index position.
1=1² , 4=2², 9=3², 16=4², 25=5²
Therefore we can also predict further terms of the given sequence as 36, 49, 64… and so on.

Some other examples of progression:

•  1,4,7,10,13,…………
•  5,1,-3,-7,-11,…….
•  5,15,45,135,405,….. 

Commonly known sequences

1. Arithmetic Sequence (or) Arithmetic Progression
2. Geometric Sequence (or) Geometric Progression
3. Harmonic Sequence (or) Harmonic Progression 
4. Fibonacci Sequence

Arithmetic Progression  (or) Arithmetic Sequence

In arithmetic progression, each term has an equal increment or decrement w.r.t to its preceding term. The change between any two consecutive terms is the same throughout the sequence.

In general, we denote the first term of this sequence by ‘a’, and the constant change in between two terms is denoted by ‘d’.

‘d’ also called common difference can be any real number.

Generally, any Arithmetic Sequence can be represented in the form:

a, a+d, a+2d, a+3d, a+4d,…………….

Example: Consider the sequence 10, 17 ,24 ,31,…..  

Solution:

We can see that common difference=17-10=24-17=31-24=7
Here 
a=10 and 
d=7.

Some other examples of arithmetic progression:

• 2,4,6,8,10,….
• 10,5,0,-5,-10,…..

 Some more generalized results for an arithmetic sequence

• In general n^th term of an Arithmetic Progression is denoted by T_n and is given by the formula

T_n= a + (n-1)d

• Sum of n terms of an Arithmetic Progression is denoted by S_n and is given by:

S_n= n(2a+(n-1)d)/2  or  S_n=n(a+a_n)/2

Note:

• In Arithmetic Progression the sum of the i^th term from the start and i^th term from the end is constant for any value of i such that 1<= i <=n.
  Example: 2,7,12,17,22,27
  In the above example 27+2=7+22=12+17=29
• If we add or subtract a constant real number from every term of an Arithmetic Progression then the resultant terms obtained are also in Arithmetic Progression with the same common difference.
• If a,b,c are in A.P then 2b=a+c.
• If we select terms after equal intervals in an Arithmetic Progression then the selected term also forms an A.P.

Geometric Progression (or) Geometric Sequence

In geometric progression, all the terms are incremented or decremented by multiplying or dividing a fixed number. The ratio of any two consecutive terms from the sequence is constant throughout. It is called the ‘common ratio’ for a Geometric Progression and is denoted by the letter ‘r’ of the English alphabet.

In general, any Geometric Progression can be represented in the form:

a, ar, ar², ar³, ar⁴,……………………

Example: Consider the sequence 1,2,4,8,16,32,……..

Solution:

If you observe , we have (2/1)=(4/2)=(8/4)=(16/8)=(32/16)=2
Here 
a=1 and 
r=2

Some other examples of arithmetic progression:

• 2,1,(0.5),(0.25),(0.125),….
• 4,12,36,108,324….. 

Some more generalized results for a Geometric Progression.

• In general n^th term of a Geometric Progression is denoted by T_n and is given by the formula

T_n = ar^(n-1)

• Sum of n terms of a Geometric Progression is denoted by S_n and is given by:

Sn=  (ar^{n- 1})/(r-1)  where r ≠1

when r=1, 

S_n simply becomes n×a.

Some interesting Properties:

1. If we multiply or divide a constant real number to each term of a Geometric Progression it still remains a geometric sequence with the same common ratio.
2. Sum of Infinite Geometric sequence when |r|<1 is given by   

 S_∞= a/(1-r)

       3. If a,b, and c are in G.P then b²=ac.

Harmonic Progression (or) Harmonic Sequence

In harmonic progression, each term when reciprocated has an equal increment or decrement w.r.t to the reciprocal of its preceding term. The change between any two reciprocated consecutive terms is the same throughout the sequence.

Note: Reciprocal terms of harmonic progression are in arithmetic progression.

In general, any Harmonic Progression can be represented in the form:

1/a, 1/(a+d)  ,1/(a+2d) ,1/(a+3d) , ……..

Example: Consider the sequence 1/5, 1/8, 1/11, 1/14, 1/17,……..

Solution:

We can clearly notice that when we reciprocate the terms it forms an arithmetic progression

5, 8, 11, 14, 17………………..

In general n^th term of a Harmonic Progression is denoted by T_n and is given by the formula

T_n= 1/(a+(n-1)d) 

Some other examples of Harmonic progression:

• 1/6, 1/12,  1/18,  1/24,  1/30
• 1/50, 1/42, 1/34, 1/26, 1/18

If a,b and c are in H.P then b=(2ac)/(a+c).

Fibonacci Sequence

A collection of numbers where every term (after the second) is the sum of the preceding two numbers.

A simple example of Fibonacci Sequence: 0,1,1,2,3,5,8,13,21,34,55,…………………

So when we declare the first two terms of the Fibonacci sequence than in general the nth term (F_n) is given by:

F_n=F_n-1+F_n-2 

Some other examples of the Fibonacci Sequence:

• 2,2,4,6,10,16,26,……..
• 1,-1,0,-1,-1,-2,-3,……

Which term of the AP 21, 18, 15 is zero?

Solution:

Lets consider the general term of an Arithmetic progression :T_n=a+(n-1)d

where ‘a’ is the first term and ‘d’ is the common difference.

Equating T_n to 0 , we have

⇒ a+(n-1)d=0

⇒ (n-1)d=-a

⇒n=1-(a/d)

As ‘n’ is natural number n≥1

Therefore n=1-(a/d)≥1

                  ⇒ -(a/d)≥0⇒   (a/d)≤0

So in order to have a term equal to 0 in a arithmetic Progression the first term and common difference must be of opposite signs.

If  a is negative then d will be positive and if a is positive then d should be negative. 

Note: This is a necessary condition and not sufficient condition as n should only be a natural number.

Now,

Which term of an AP 21, 18, 15 is zero?

Here,

a=21 and d=-3 (‘a’ and ‘d’ are of opposite signs so we can proceed further)

T_n = a+(n-1)d

0 = 21+(n-1)(-3)

⇒ n=8

8th term of the given A.P will be zero.

Sample Problems

Problem 1: Which term of the given A.P 100, 96, 92, 88,…… is 0?

Solution:

Here 

a=100 and d=-4 (‘a’ and ‘d’ are of opposite signs so we can proceed further)

T_n=a+(n-1)d

0 = 100+(n-1)(-4)

⇒ n=26

26th term of the given A.P will be zero.

Problem 2: Which term of the given A.P  -180, -135, -90…… is 0?

Solution:

Here 

a=-180 and d=45 (‘a’ and ‘d’ are of opposite signs so we can proceed further)

T_n=a+(n-1)d

0 = -180+(n-1)(45)

⇒ n=5

5^th term of the given A.P will be zero

Problem 3: Which term of the given A.P  2, 9, 16, 23…… is 0?

Solution:

Here 

a=2 and d=7 (‘a’ and ‘d’ are of the same signs so we don’t need to proceed further)

But still if we check 

T_n=a+(n-1)d

0 = 2+(n-1)(7)

⇒ n=(5/7) which is not a natural number so there is no term in the given A.P with value as 0.

Problem 4: Which term of the given A.P  50,44,38,…………?

Solution:

Here 

a=50 and d=-6 (a and d are of opposite signs so we can proceed further)

T_n=a+(n-1)d

0 = 50+(n-1)(-6)

⇒ n=(28/3) which is not a natural number or a defined index position in the A.P. So there is no term in this A.P that is 0.

Hence we can see that even when a and d are of opposite signs n cannot be guaranteed as a natural number.