Question:
Find the average of odd numbers from 7 to 189
Correct Answer
98
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 189
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 189 are
7, 9, 11, . . . . 189
After observing the above list of the odd numbers from 7 to 189 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 189 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 189
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 189
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the odd numbers from 7 to 189
= 7 + 189/2
= 196/2 = 98
Thus, the average of the odd numbers from 7 to 189 = 98 Answer
Method (2) to find the average of the odd numbers from 7 to 189
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 189 are
7, 9, 11, . . . . 189
The odd numbers from 7 to 189 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 189
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the odd numbers from 7 to 189
189 = 7 + (n – 1) × 2
⇒ 189 = 7 + 2 n – 2
⇒ 189 = 7 – 2 + 2 n
⇒ 189 = 5 + 2 n
After transposing 5 to LHS
⇒ 189 – 5 = 2 n
⇒ 184 = 2 n
After rearranging the above expression
⇒ 2 n = 184
After transposing 2 to RHS
⇒ n = 184/2
⇒ n = 92
Thus, the number of terms of odd numbers from 7 to 189 = 92
This means 189 is the 92th term.
Finding the sum of the given odd numbers from 7 to 189
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given odd numbers from 7 to 189
= 92/2 (7 + 189)
= 92/2 × 196
= 92 × 196/2
= 18032/2 = 9016
Thus, the sum of all terms of the given odd numbers from 7 to 189 = 9016
And, the total number of terms = 92
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given odd numbers from 7 to 189
= 9016/92 = 98
Thus, the average of the given odd numbers from 7 to 189 = 98 Answer
Similar Questions
(1) Find the average of the first 3100 odd numbers.
(2) What is the average of the first 544 even numbers?
(3) Find the average of the first 2424 odd numbers.
(4) Find the average of the first 3943 even numbers.
(5) Find the average of the first 4411 even numbers.
(6) Find the average of odd numbers from 15 to 1315
(7) What will be the average of the first 4877 odd numbers?
(8) Find the average of the first 3309 even numbers.
(9) What will be the average of the first 4863 odd numbers?
(10) What is the average of the first 668 even numbers?