Consecutive odd numbers – Solved Examples

Consecutive odd numbers

Any number in the form of (2n + 1), where n is any positive integer is termed as an odd number. 2 is not a factor of these numbers.

Also for negative odd numbers, it can be written as (2n – 1) where n is any negative integer.

The series of odd numbers in which every number is followed by its successive odd number are known as consecutive odd numbers. The series of odd numbers are arranged in ascending order.

The difference between any two consecutive numbers of this series is equal to 2.

Examples

1. 3, 5, 7 and so on.
2. 17, 19, 21 and so on.
3. -3, -1, 1, 3, and so on.

17, 21 and 23 are not odd consecutive numbers because there is an odd number 19 in between 17 and 21.

If the first number in this series is of the form (2a + 1), the next numbers of these series are (2a + 3), (2a + 5), ( 2a + 7),( 2a + 9) and so on. Here “a” is a positive integer.

If the first number in this series is of the form (2a – 1), the next numbers of these series are (2a + 1), (2a + 3), (2a + 5), ( 2a + 7), ( 2a + 9) and so on. Here “a” is a negative integer.

If there are an odd number of terms in the series of odd consecutive numbers, then the sum of all the terms is odd.

If there are an even number of terms in the series of odd consecutive numbers, then the sum of all the terms is even.

Solved Examples

1. Given below is a series of odd numbers-

-55, -53, -51 and so on. Find the next 5 numbers of this series.

For obtaining the next 5 numbers of the series we have to keep adding 2 to the previous

(-51) + 2 = (-49)

(-49) + 2 = (-47)

(-47) + 2 = (-45)

(-45) + 2 = (-43)

(-43) + 2 = (-41)

Hence the next five numbers of the given series are -49, -47, -45, -43 and -41.

2. The sum of two odd numbers is equal to 92. The difference between these numbers is equal to 2. Find them.

Let one of the numbers be (2a + 1). 

Next odd number = ( 2a + 1) +2 = (2a + 3)

Sum = 92(Given)

⇒ (2a + 1) + (2a + 3) = 92

⇒ 4a + 4 = 92

⇒ 4a = 92 – 4 = 88

⇒ a = \frac{88}{4} = 22

∴ 2a + 1 = (2 × 22) + 1 = 45

And, 2a + 3 = (2 × 22) + 3 = 47

Check 

45 + 47 = 92

Hence the two numbers are 44 and 47.

3. The product of two odd numbers is equal to 63. They are consecutive. Find them.

Let one of the numbers be (2a + 1).

Next odd number = (2a + 1) + 2 = (2a + 3).

Product = (2a+1) (2a+3)

               = 4 a^{2}+8 a+3

Now, Product = 63(Given)

\Rightarrow \quad 4 a^{2}+8 a+3=63

\Rightarrow \quad 4 a^{2}+8 a=63-3=60
\Rightarrow \quad 4\left(a^{2}+2 a\right)=60
\Rightarrow \quad\left(a^{2}+2 a\right)=\frac{60} {4}=15
\Rightarrow \quad a(a+2)=15
\Rightarrow \quad a=3

2a + 1 = (2 × 3) + 1 

⇒ 2a + 1 = 7

2a + 3 = (2 × 3) + 3 

⇒ 2a + 3 = 9

Check

7 × 9 = 63

Therefore the required two numbers are 7 and 9.