# Which term of an AP is zero?

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 …**

**N ^{th} term of an AP**

An AP whose first term is a and the common difference is d. Then the n^{th} term of the AP is given by –

**a _{n} = 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 n

^{th}term of the AP is given by –a

_{n}= a + ( n – 1 ) dHere, a

_{n}= 0 since it is given that the n^{th}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 32 ^{th} term of the AP.**

**Solution :**

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

10 – 7 = 3 (2

^{nd}term – 1^{st}term)13 – 10 = 3 (3

^{rd}term – 2^{nd}term)16 – 13 = 3 (4

^{th}term – 3^{rd}term)19 – 16 = 3 (5

^{th}term – 4^{th}term)The first term (a) of the AP is 7 and the common difference( d ) of the AP is 3.

The formula for 32

^{th}term of the AP is –a

_{n}= a + ( n – 1 ) da

_{32}= 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, a

_{n}= 0 , d = -2 , a = 10. So,By using the formula for the nth term of the AP –

a

_{n}= 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, a

_{n}= 0a

_{n}= a + (n – 1) d=> 0 = 27 + (n – 1)×(-3)

=> -27 = (n – 1)× (-3)

=> n – 1 = 9

=> n = 9 + 1 = 10