Question:
Find the average of odd numbers from 5 to 1067
Correct Answer
536
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 1067
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 1067 are
5, 7, 9, . . . . 1067
After observing the above list of the odd numbers from 5 to 1067 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 1067 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 1067
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 1067
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 5 to 1067
= 5 + 1067/2
= 1072/2 = 536
Thus, the average of the odd numbers from 5 to 1067 = 536 Answer
Method (2) to find the average of the odd numbers from 5 to 1067
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 1067 are
5, 7, 9, . . . . 1067
The odd numbers from 5 to 1067 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 1067
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 5 to 1067
1067 = 5 + (n – 1) × 2
⇒ 1067 = 5 + 2 n – 2
⇒ 1067 = 5 – 2 + 2 n
⇒ 1067 = 3 + 2 n
After transposing 3 to LHS
⇒ 1067 – 3 = 2 n
⇒ 1064 = 2 n
After rearranging the above expression
⇒ 2 n = 1064
After transposing 2 to RHS
⇒ n = 1064/2
⇒ n = 532
Thus, the number of terms of odd numbers from 5 to 1067 = 532
This means 1067 is the 532th term.
Finding the sum of the given odd numbers from 5 to 1067
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 5 to 1067
= 532/2 (5 + 1067)
= 532/2 × 1072
= 532 × 1072/2
= 570304/2 = 285152
Thus, the sum of all terms of the given odd numbers from 5 to 1067 = 285152
And, the total number of terms = 532
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 5 to 1067
= 285152/532 = 536
Thus, the average of the given odd numbers from 5 to 1067 = 536 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 546
(2) Find the average of even numbers from 4 to 1988
(3) Find the average of the first 1055 odd numbers.
(4) Find the average of the first 2044 even numbers.
(5) Find the average of the first 3243 odd numbers.
(6) Find the average of odd numbers from 15 to 1447
(7) Find the average of even numbers from 12 to 458
(8) Find the average of the first 4176 even numbers.
(9) Find the average of the first 3497 even numbers.
(10) Find the average of the first 4822 even numbers.