# Which term of an AP is zero?

• Last Updated : 17 Aug, 2021

When a group of numbers is arranged in such an order that it exhibits a pattern, then the numbers are said to be in Progression. It is also known as Sequence or series. In Mathematics, there are mainly three types of Progression – Arithmetic Progression, Geometric Progression, and Harmonic Progression. Let’s learn more in detail about Arithmetic progression,

### Arithmetic Progression

When the difference between any two consecutive terms of a sequence of numbers is always the same, then the sequence is said to be in Arithmetic Progression. Examples, Sequence: 10, 20, 30, 40 – This sequence is in AP because the difference between any two consecutive terms is same (20 – 10 = 30 – 20 = 40 – 30 = 10). Sequence: 0, 2, 4, 6. This sequence is also in AP as the difference between any two consecutive terms is same (2 – 0 = 4 – 2 = 6 – 4 = 2)

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First-term of an AP

The first term of an AP is denoted by a. It is the first term to exist in the sequence. Examples: Sequence : 3, 5, 7, 9, 11- This sequence is in AP and the difference between any two consecutive terms is 2. The first term of the AP is 3. Sequence : 15, 21, 27, 33- The sequence is in AP as the difference between any two consecutive terms is 6. The first term of the AP is 15.

Common Difference of an AP

In AP, the difference between the consecutive terms is a constant. This constant is called as the common difference of the AP. It is denoted by d. An AP whose first term is a and the common difference is d can be written as –

a, a+d, a+2d, a+3d, a+4d, a+5d …

Nth term of an AP

An AP whose first term is a and the common difference is d. Then the nth term of the AP is given by –

an = a + (n – 1) d

### Which term of the AP is zero ?

Given an AP, we have to calculate the position of the term whose value is zero. Let a be the first term of the AP and d be the common difference of the AP,

The nth term of the AP is given by –

an = a + ( n – 1 ) d

Here, an = 0 since it is given that the nth term of the AP is zero. So,

0 = a + ( n – 1 ) d

a + ( n – 1 ) d = 0

( n – 1 ) d = -a

n – 1  = -a / d

n = -a / d + 1

n = 1 – a / d

The term of the AP whose value is zero is given by : n = 1 – a / d

### Sample Problem

Question`1: Given a sequence :  7, 10, 13, 16, 19… Find whether the sequence is in AP or not. If it is in AP, find the 32th term of the AP.

Solution :

The above sequence is in AP with a common difference of 3.

10 – 7 = 3 (2nd term – 1st term)

13 – 10 = 3 (3rd term – 2nd term)

16 – 13 = 3 (4th term – 3rd term)

19 – 16 = 3 (5th term – 4th term)

The first term (a) of the AP is 7 and the common difference( d ) of the AP is 3.

The formula for 32th term of the AP is –

an = a + ( n – 1 ) d

a32 = 7 + ( 32 – 1)* 3

= 7 + 31 * 3

= 7 + 93

= 100

Question 2: Given an AP : 10, 8, 6, 4… Which term of the given AP is zero ?

Solution :

Here, the first term ( a ) of the AP is 10 and the common difference (d) = 8 – 10 = -2.

Method 1 :

By using formula for the nth term of AP which is zero, we have-

=> n = 1 – 10 /-2

=> n = 1 – (-5)

=> n = 1 + 5 = 6

Method 2 :

Here, an = 0 , d = -2 , a = 10. So,

By using the formula for the nth term of the AP –

an = a + (n – 1) d

=> 0 = 10 + (n – 1)× (-2)

=> -10 = (n – 1)× (-2)

=> 5 = n – 1

=> n – 1 = 5

=> n = 5 + 1 = 6

Question 3: Given an AP : 27, 24, 21, 18…..  Which term of the given AP is zero ?

Solution :

The first term of the AP (a) = 27 and the common difference (d) is 24 – 27 = -3

We have, an = 0

an = a + (n – 1) d

=> 0 = 27 + (n – 1)×(-3)

=> -27 = (n – 1)× (-3)

=> n – 1 = 9

=> n = 9 + 1 = 10

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