Question:
Find the average of odd numbers from 3 to 1431
Correct Answer
717
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 1431
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 1431 are
3, 5, 7, . . . . 1431
After observing the above list of the odd numbers from 3 to 1431 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 1431 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 1431
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1431
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 1431
= 3 + 1431/2
= 1434/2 = 717
Thus, the average of the odd numbers from 3 to 1431 = 717 Answer
Method (2) to find the average of the odd numbers from 3 to 1431
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 1431 are
3, 5, 7, . . . . 1431
The odd numbers from 3 to 1431 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1431
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 1431
1431 = 3 + (n – 1) × 2
⇒ 1431 = 3 + 2 n – 2
⇒ 1431 = 3 – 2 + 2 n
⇒ 1431 = 1 + 2 n
After transposing 1 to LHS
⇒ 1431 – 1 = 2 n
⇒ 1430 = 2 n
After rearranging the above expression
⇒ 2 n = 1430
After transposing 2 to RHS
⇒ n = 1430/2
⇒ n = 715
Thus, the number of terms of odd numbers from 3 to 1431 = 715
This means 1431 is the 715th term.
Finding the sum of the given odd numbers from 3 to 1431
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 1431
= 715/2 (3 + 1431)
= 715/2 × 1434
= 715 × 1434/2
= 1025310/2 = 512655
Thus, the sum of all terms of the given odd numbers from 3 to 1431 = 512655
And, the total number of terms = 715
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 1431
= 512655/715 = 717
Thus, the average of the given odd numbers from 3 to 1431 = 717 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 763
(2) Find the average of the first 4814 even numbers.
(3) What is the average of the first 99 even numbers?
(4) What will be the average of the first 4889 odd numbers?
(5) Find the average of even numbers from 4 to 198
(6) Find the average of the first 482 odd numbers.
(7) What is the average of the first 740 even numbers?
(8) Find the average of even numbers from 12 to 1924
(9) What is the average of the first 696 even numbers?
(10) What is the average of the first 96 odd numbers?