Question:
Find the average of odd numbers from 9 to 1441
Correct Answer
725
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 1441
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 1441 are
9, 11, 13, . . . . 1441
After observing the above list of the odd numbers from 9 to 1441 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 1441 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 1441
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 1441
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 1441
= 9 + 1441/2
= 1450/2 = 725
Thus, the average of the odd numbers from 9 to 1441 = 725 Answer
Method (2) to find the average of the odd numbers from 9 to 1441
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 1441 are
9, 11, 13, . . . . 1441
The odd numbers from 9 to 1441 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 1441
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 1441
1441 = 9 + (n – 1) × 2
⇒ 1441 = 9 + 2 n – 2
⇒ 1441 = 9 – 2 + 2 n
⇒ 1441 = 7 + 2 n
After transposing 7 to LHS
⇒ 1441 – 7 = 2 n
⇒ 1434 = 2 n
After rearranging the above expression
⇒ 2 n = 1434
After transposing 2 to RHS
⇒ n = 1434/2
⇒ n = 717
Thus, the number of terms of odd numbers from 9 to 1441 = 717
This means 1441 is the 717th term.
Finding the sum of the given odd numbers from 9 to 1441
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 1441
= 717/2 (9 + 1441)
= 717/2 × 1450
= 717 × 1450/2
= 1039650/2 = 519825
Thus, the sum of all terms of the given odd numbers from 9 to 1441 = 519825
And, the total number of terms = 717
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 1441
= 519825/717 = 725
Thus, the average of the given odd numbers from 9 to 1441 = 725 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1962
(2) Find the average of even numbers from 10 to 1966
(3) Find the average of the first 3679 odd numbers.
(4) Find the average of the first 1668 odd numbers.
(5) Find the average of even numbers from 10 to 1958
(6) Find the average of odd numbers from 7 to 449
(7) What will be the average of the first 4262 odd numbers?
(8) Find the average of even numbers from 10 to 1354
(9) Find the average of odd numbers from 3 to 259
(10) Find the average of odd numbers from 9 to 269