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

Zero(0) is the 8th term of the Arithmetic Progression (AP) 21, 18, 15,... and the detailed solution for the same is added below:

Which Term of 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.

Similar 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.