Question : Find the average of odd numbers from 15 to 1795
Correct Answer 905
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1795
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1795 are
15, 17, 19, . . . . 1795
After observing the above list of the odd numbers from 15 to 1795 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1795 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1795
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1795
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 1795
= 15 + 1795/2
= 1810/2 = 905
Thus, the average of the odd numbers from 15 to 1795 = 905 Answer
Method (2) to find the average of the odd numbers from 15 to 1795
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1795 are
15, 17, 19, . . . . 1795
The odd numbers from 15 to 1795 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1795
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 1795
1795 = 15 + (n – 1) × 2
⇒ 1795 = 15 + 2 n – 2
⇒ 1795 = 15 – 2 + 2 n
⇒ 1795 = 13 + 2 n
After transposing 13 to LHS
⇒ 1795 – 13 = 2 n
⇒ 1782 = 2 n
After rearranging the above expression
⇒ 2 n = 1782
After transposing 2 to RHS
⇒ n = 1782/2
⇒ n = 891
Thus, the number of terms of odd numbers from 15 to 1795 = 891
This means 1795 is the 891th term.
Finding the sum of the given odd numbers from 15 to 1795
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 1795
= 891/2 (15 + 1795)
= 891/2 × 1810
= 891 × 1810/2
= 1612710/2 = 806355
Thus, the sum of all terms of the given odd numbers from 15 to 1795 = 806355
And, the total number of terms = 891
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 1795
= 806355/891 = 905
Thus, the average of the given odd numbers from 15 to 1795 = 905 Answer
Similar Questions
(1) Find the average of odd numbers from 11 to 229
(2) Find the average of even numbers from 12 to 836
(3) Find the average of the first 867 odd numbers.
(4) What is the average of the first 1583 even numbers?
(5) What is the average of the first 1601 even numbers?
(6) Find the average of even numbers from 4 to 606
(7) Find the average of odd numbers from 3 to 753
(8) Find the average of odd numbers from 5 to 651
(9) Find the average of the first 3667 odd numbers.
(10) What is the average of the first 129 even numbers?