Question:
Find the average of odd numbers from 15 to 1519
Correct Answer
767
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1519
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1519 are
15, 17, 19, . . . . 1519
After observing the above list of the odd numbers from 15 to 1519 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1519 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1519
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1519
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 15 to 1519
= 15 + 1519/2
= 1534/2 = 767
Thus, the average of the odd numbers from 15 to 1519 = 767 Answer
Method (2) to find the average of the odd numbers from 15 to 1519
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1519 are
15, 17, 19, . . . . 1519
The odd numbers from 15 to 1519 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1519
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 15 to 1519
1519 = 15 + (n – 1) × 2
⇒ 1519 = 15 + 2 n – 2
⇒ 1519 = 15 – 2 + 2 n
⇒ 1519 = 13 + 2 n
After transposing 13 to LHS
⇒ 1519 – 13 = 2 n
⇒ 1506 = 2 n
After rearranging the above expression
⇒ 2 n = 1506
After transposing 2 to RHS
⇒ n = 1506/2
⇒ n = 753
Thus, the number of terms of odd numbers from 15 to 1519 = 753
This means 1519 is the 753th term.
Finding the sum of the given odd numbers from 15 to 1519
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 15 to 1519
= 753/2 (15 + 1519)
= 753/2 × 1534
= 753 × 1534/2
= 1155102/2 = 577551
Thus, the sum of all terms of the given odd numbers from 15 to 1519 = 577551
And, the total number of terms = 753
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 15 to 1519
= 577551/753 = 767
Thus, the average of the given odd numbers from 15 to 1519 = 767 Answer
Similar Questions
(1) Find the average of the first 1523 odd numbers.
(2) Find the average of the first 3802 odd numbers.
(3) Find the average of the first 1047 odd numbers.
(4) Find the average of the first 2390 even numbers.
(5) Find the average of odd numbers from 15 to 1611
(6) Find the average of even numbers from 4 to 914
(7) Find the average of odd numbers from 5 to 1237
(8) Find the average of the first 2439 even numbers.
(9) What will be the average of the first 4674 odd numbers?
(10) Find the average of the first 3686 even numbers.