Question:
Find the average of odd numbers from 5 to 913
Correct Answer
459
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 913
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 913 are
5, 7, 9, . . . . 913
After observing the above list of the odd numbers from 5 to 913 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 913 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 913
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 913
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 913
= 5 + 913/2
= 918/2 = 459
Thus, the average of the odd numbers from 5 to 913 = 459 Answer
Method (2) to find the average of the odd numbers from 5 to 913
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 913 are
5, 7, 9, . . . . 913
The odd numbers from 5 to 913 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 913
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 913
913 = 5 + (n – 1) × 2
⇒ 913 = 5 + 2 n – 2
⇒ 913 = 5 – 2 + 2 n
⇒ 913 = 3 + 2 n
After transposing 3 to LHS
⇒ 913 – 3 = 2 n
⇒ 910 = 2 n
After rearranging the above expression
⇒ 2 n = 910
After transposing 2 to RHS
⇒ n = 910/2
⇒ n = 455
Thus, the number of terms of odd numbers from 5 to 913 = 455
This means 913 is the 455th term.
Finding the sum of the given odd numbers from 5 to 913
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 913
= 455/2 (5 + 913)
= 455/2 × 918
= 455 × 918/2
= 417690/2 = 208845
Thus, the sum of all terms of the given odd numbers from 5 to 913 = 208845
And, the total number of terms = 455
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 913
= 208845/455 = 459
Thus, the average of the given odd numbers from 5 to 913 = 459 Answer
Similar Questions
(1) Find the average of the first 1927 odd numbers.
(2) Find the average of the first 2816 even numbers.
(3) Find the average of even numbers from 4 to 1924
(4) Find the average of even numbers from 4 to 806
(5) Find the average of odd numbers from 5 to 1267
(6) Find the average of the first 1013 odd numbers.
(7) Find the average of the first 2787 odd numbers.
(8) What is the average of the first 1465 even numbers?
(9) Find the average of the first 4809 even numbers.
(10) Find the average of even numbers from 4 to 1126