Question:
Find the average of odd numbers from 3 to 937
Correct Answer
470
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 937
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 937 are
3, 5, 7, . . . . 937
After observing the above list of the odd numbers from 3 to 937 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 937 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 937
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 937
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 937
= 3 + 937/2
= 940/2 = 470
Thus, the average of the odd numbers from 3 to 937 = 470 Answer
Method (2) to find the average of the odd numbers from 3 to 937
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 937 are
3, 5, 7, . . . . 937
The odd numbers from 3 to 937 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 937
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 937
937 = 3 + (n – 1) × 2
⇒ 937 = 3 + 2 n – 2
⇒ 937 = 3 – 2 + 2 n
⇒ 937 = 1 + 2 n
After transposing 1 to LHS
⇒ 937 – 1 = 2 n
⇒ 936 = 2 n
After rearranging the above expression
⇒ 2 n = 936
After transposing 2 to RHS
⇒ n = 936/2
⇒ n = 468
Thus, the number of terms of odd numbers from 3 to 937 = 468
This means 937 is the 468th term.
Finding the sum of the given odd numbers from 3 to 937
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 937
= 468/2 (3 + 937)
= 468/2 × 940
= 468 × 940/2
= 439920/2 = 219960
Thus, the sum of all terms of the given odd numbers from 3 to 937 = 219960
And, the total number of terms = 468
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 937
= 219960/468 = 470
Thus, the average of the given odd numbers from 3 to 937 = 470 Answer
Similar Questions
(1) Find the average of the first 906 odd numbers.
(2) Find the average of the first 2916 odd numbers.
(3) Find the average of the first 2495 even numbers.
(4) Find the average of even numbers from 4 to 158
(5) Find the average of even numbers from 10 to 838
(6) Find the average of the first 3339 even numbers.
(7) Find the average of the first 2699 even numbers.
(8) Find the average of even numbers from 6 to 710
(9) Find the average of the first 2665 even numbers.
(10) Find the average of even numbers from 10 to 122