Question : Find the average of odd numbers from 3 to 737
Correct Answer 370
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 737
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 737 are
3, 5, 7, . . . . 737
After observing the above list of the odd numbers from 3 to 737 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 737 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 737
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 737
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 3 to 737
= 3 + 737/2
= 740/2 = 370
Thus, the average of the odd numbers from 3 to 737 = 370 Answer
Method (2) to find the average of the odd numbers from 3 to 737
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 737 are
3, 5, 7, . . . . 737
The odd numbers from 3 to 737 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 737
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 3 to 737
737 = 3 + (n – 1) × 2
⇒ 737 = 3 + 2 n – 2
⇒ 737 = 3 – 2 + 2 n
⇒ 737 = 1 + 2 n
After transposing 1 to LHS
⇒ 737 – 1 = 2 n
⇒ 736 = 2 n
After rearranging the above expression
⇒ 2 n = 736
After transposing 2 to RHS
⇒ n = 736/2
⇒ n = 368
Thus, the number of terms of odd numbers from 3 to 737 = 368
This means 737 is the 368th term.
Finding the sum of the given odd numbers from 3 to 737
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 3 to 737
= 368/2 (3 + 737)
= 368/2 × 740
= 368 × 740/2
= 272320/2 = 136160
Thus, the sum of all terms of the given odd numbers from 3 to 737 = 136160
And, the total number of terms = 368
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 3 to 737
= 136160/368 = 370
Thus, the average of the given odd numbers from 3 to 737 = 370 Answer
Similar Questions
(1) Find the average of the first 2284 odd numbers.
(2) What will be the average of the first 4472 odd numbers?
(3) Find the average of the first 929 odd numbers.
(4) What is the average of the first 1575 even numbers?
(5) Find the average of odd numbers from 9 to 151
(6) Find the average of the first 4486 even numbers.
(7) Find the average of odd numbers from 3 to 637
(8) What is the average of the first 965 even numbers?
(9) Find the average of the first 3286 odd numbers.
(10) What is the average of the first 109 even numbers?