Question:
Find the average of odd numbers from 7 to 727
Correct Answer
367
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 727
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 727 are
7, 9, 11, . . . . 727
After observing the above list of the odd numbers from 7 to 727 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 727 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 727
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 727
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 727
= 7 + 727/2
= 734/2 = 367
Thus, the average of the odd numbers from 7 to 727 = 367 Answer
Method (2) to find the average of the odd numbers from 7 to 727
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 727 are
7, 9, 11, . . . . 727
The odd numbers from 7 to 727 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 727
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 727
727 = 7 + (n – 1) × 2
⇒ 727 = 7 + 2 n – 2
⇒ 727 = 7 – 2 + 2 n
⇒ 727 = 5 + 2 n
After transposing 5 to LHS
⇒ 727 – 5 = 2 n
⇒ 722 = 2 n
After rearranging the above expression
⇒ 2 n = 722
After transposing 2 to RHS
⇒ n = 722/2
⇒ n = 361
Thus, the number of terms of odd numbers from 7 to 727 = 361
This means 727 is the 361th term.
Finding the sum of the given odd numbers from 7 to 727
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 727
= 361/2 (7 + 727)
= 361/2 × 734
= 361 × 734/2
= 264974/2 = 132487
Thus, the sum of all terms of the given odd numbers from 7 to 727 = 132487
And, the total number of terms = 361
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 727
= 132487/361 = 367
Thus, the average of the given odd numbers from 7 to 727 = 367 Answer
Similar Questions
(1) Find the average of the first 382 odd numbers.
(2) Find the average of the first 3771 even numbers.
(3) Find the average of odd numbers from 11 to 1375
(4) What is the average of the first 443 even numbers?
(5) Find the average of even numbers from 12 to 350
(6) Find the average of odd numbers from 3 to 1215
(7) What is the average of the first 111 odd numbers?
(8) Find the average of the first 4304 even numbers.
(9) Find the average of odd numbers from 13 to 71
(10) Find the average of odd numbers from 5 to 1211