Question : Find the average of odd numbers from 3 to 819
Correct Answer 411
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 819
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 819 are
3, 5, 7, . . . . 819
After observing the above list of the odd numbers from 3 to 819 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 819 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 819
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 819
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 819
= 3 + 819/2
= 822/2 = 411
Thus, the average of the odd numbers from 3 to 819 = 411 Answer
Method (2) to find the average of the odd numbers from 3 to 819
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 819 are
3, 5, 7, . . . . 819
The odd numbers from 3 to 819 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 819
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 819
819 = 3 + (n – 1) × 2
⇒ 819 = 3 + 2 n – 2
⇒ 819 = 3 – 2 + 2 n
⇒ 819 = 1 + 2 n
After transposing 1 to LHS
⇒ 819 – 1 = 2 n
⇒ 818 = 2 n
After rearranging the above expression
⇒ 2 n = 818
After transposing 2 to RHS
⇒ n = 818/2
⇒ n = 409
Thus, the number of terms of odd numbers from 3 to 819 = 409
This means 819 is the 409th term.
Finding the sum of the given odd numbers from 3 to 819
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 819
= 409/2 (3 + 819)
= 409/2 × 822
= 409 × 822/2
= 336198/2 = 168099
Thus, the sum of all terms of the given odd numbers from 3 to 819 = 168099
And, the total number of terms = 409
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 819
= 168099/409 = 411
Thus, the average of the given odd numbers from 3 to 819 = 411 Answer
Similar Questions
(1) What is the average of the first 190 odd numbers?
(2) Find the average of odd numbers from 5 to 71
(3) Find the average of the first 1372 odd numbers.
(4) Find the average of odd numbers from 3 to 1457
(5) Find the average of even numbers from 12 to 1174
(6) What is the average of the first 672 even numbers?
(7) Find the average of odd numbers from 11 to 735
(8) Find the average of the first 3564 even numbers.
(9) Find the average of the first 4441 even numbers.
(10) Find the average of odd numbers from 3 to 629