Question : Find the average of odd numbers from 15 to 1779
Correct Answer 897
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1779
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1779 are
15, 17, 19, . . . . 1779
After observing the above list of the odd numbers from 15 to 1779 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1779 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1779
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1779
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 1779
= 15 + 1779/2
= 1794/2 = 897
Thus, the average of the odd numbers from 15 to 1779 = 897 Answer
Method (2) to find the average of the odd numbers from 15 to 1779
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1779 are
15, 17, 19, . . . . 1779
The odd numbers from 15 to 1779 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1779
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 1779
1779 = 15 + (n – 1) × 2
⇒ 1779 = 15 + 2 n – 2
⇒ 1779 = 15 – 2 + 2 n
⇒ 1779 = 13 + 2 n
After transposing 13 to LHS
⇒ 1779 – 13 = 2 n
⇒ 1766 = 2 n
After rearranging the above expression
⇒ 2 n = 1766
After transposing 2 to RHS
⇒ n = 1766/2
⇒ n = 883
Thus, the number of terms of odd numbers from 15 to 1779 = 883
This means 1779 is the 883th term.
Finding the sum of the given odd numbers from 15 to 1779
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 1779
= 883/2 (15 + 1779)
= 883/2 × 1794
= 883 × 1794/2
= 1584102/2 = 792051
Thus, the sum of all terms of the given odd numbers from 15 to 1779 = 792051
And, the total number of terms = 883
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 1779
= 792051/883 = 897
Thus, the average of the given odd numbers from 15 to 1779 = 897 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 1311
(2) Find the average of even numbers from 12 to 988
(3) Find the average of the first 3679 odd numbers.
(4) Find the average of odd numbers from 13 to 1347
(5) Find the average of odd numbers from 9 to 729
(6) What is the average of the first 138 odd numbers?
(7) Find the average of the first 3974 even numbers.
(8) Find the average of the first 791 odd numbers.
(9) What is the average of the first 155 even numbers?
(10) Find the average of the first 2175 even numbers.