Question:
Find the average of odd numbers from 15 to 1705
Correct Answer
860
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1705
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1705 are
15, 17, 19, . . . . 1705
After observing the above list of the odd numbers from 15 to 1705 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1705 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1705
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1705
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 1705
= 15 + 1705/2
= 1720/2 = 860
Thus, the average of the odd numbers from 15 to 1705 = 860 Answer
Method (2) to find the average of the odd numbers from 15 to 1705
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1705 are
15, 17, 19, . . . . 1705
The odd numbers from 15 to 1705 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1705
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 1705
1705 = 15 + (n – 1) × 2
⇒ 1705 = 15 + 2 n – 2
⇒ 1705 = 15 – 2 + 2 n
⇒ 1705 = 13 + 2 n
After transposing 13 to LHS
⇒ 1705 – 13 = 2 n
⇒ 1692 = 2 n
After rearranging the above expression
⇒ 2 n = 1692
After transposing 2 to RHS
⇒ n = 1692/2
⇒ n = 846
Thus, the number of terms of odd numbers from 15 to 1705 = 846
This means 1705 is the 846th term.
Finding the sum of the given odd numbers from 15 to 1705
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 1705
= 846/2 (15 + 1705)
= 846/2 × 1720
= 846 × 1720/2
= 1455120/2 = 727560
Thus, the sum of all terms of the given odd numbers from 15 to 1705 = 727560
And, the total number of terms = 846
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 1705
= 727560/846 = 860
Thus, the average of the given odd numbers from 15 to 1705 = 860 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1036
(2) Find the average of even numbers from 6 to 1230
(3) Find the average of the first 2624 even numbers.
(4) What is the average of the first 634 even numbers?
(5) Find the average of odd numbers from 3 to 899
(6) What is the average of the first 750 even numbers?
(7) Find the average of even numbers from 6 to 1660
(8) What is the average of the first 1551 even numbers?
(9) Find the average of the first 3770 odd numbers.
(10) What is the average of the first 1121 even numbers?