Question:
Find the average of odd numbers from 9 to 773
Correct Answer
391
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 773
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 773 are
9, 11, 13, . . . . 773
After observing the above list of the odd numbers from 9 to 773 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 773 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 773
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 773
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 9 to 773
= 9 + 773/2
= 782/2 = 391
Thus, the average of the odd numbers from 9 to 773 = 391 Answer
Method (2) to find the average of the odd numbers from 9 to 773
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 773 are
9, 11, 13, . . . . 773
The odd numbers from 9 to 773 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 773
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 9 to 773
773 = 9 + (n – 1) × 2
⇒ 773 = 9 + 2 n – 2
⇒ 773 = 9 – 2 + 2 n
⇒ 773 = 7 + 2 n
After transposing 7 to LHS
⇒ 773 – 7 = 2 n
⇒ 766 = 2 n
After rearranging the above expression
⇒ 2 n = 766
After transposing 2 to RHS
⇒ n = 766/2
⇒ n = 383
Thus, the number of terms of odd numbers from 9 to 773 = 383
This means 773 is the 383th term.
Finding the sum of the given odd numbers from 9 to 773
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 9 to 773
= 383/2 (9 + 773)
= 383/2 × 782
= 383 × 782/2
= 299506/2 = 149753
Thus, the sum of all terms of the given odd numbers from 9 to 773 = 149753
And, the total number of terms = 383
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 9 to 773
= 149753/383 = 391
Thus, the average of the given odd numbers from 9 to 773 = 391 Answer
Similar Questions
(1) Find the average of the first 707 odd numbers.
(2) Find the average of odd numbers from 11 to 113
(3) Find the average of even numbers from 8 to 1128
(4) Find the average of even numbers from 10 to 1746
(5) Find the average of even numbers from 10 to 1726
(6) Find the average of odd numbers from 9 to 717
(7) Find the average of the first 4499 even numbers.
(8) Find the average of odd numbers from 15 to 1119
(9) Find the average of odd numbers from 5 to 1385
(10) Find the average of odd numbers from 3 to 623