Question:
Find the average of odd numbers from 11 to 951
Correct Answer
481
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 951
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 951 are
11, 13, 15, . . . . 951
After observing the above list of the odd numbers from 11 to 951 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 951 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 951
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 951
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 11 to 951
= 11 + 951/2
= 962/2 = 481
Thus, the average of the odd numbers from 11 to 951 = 481 Answer
Method (2) to find the average of the odd numbers from 11 to 951
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 951 are
11, 13, 15, . . . . 951
The odd numbers from 11 to 951 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 951
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 11 to 951
951 = 11 + (n – 1) × 2
⇒ 951 = 11 + 2 n – 2
⇒ 951 = 11 – 2 + 2 n
⇒ 951 = 9 + 2 n
After transposing 9 to LHS
⇒ 951 – 9 = 2 n
⇒ 942 = 2 n
After rearranging the above expression
⇒ 2 n = 942
After transposing 2 to RHS
⇒ n = 942/2
⇒ n = 471
Thus, the number of terms of odd numbers from 11 to 951 = 471
This means 951 is the 471th term.
Finding the sum of the given odd numbers from 11 to 951
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 11 to 951
= 471/2 (11 + 951)
= 471/2 × 962
= 471 × 962/2
= 453102/2 = 226551
Thus, the sum of all terms of the given odd numbers from 11 to 951 = 226551
And, the total number of terms = 471
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 11 to 951
= 226551/471 = 481
Thus, the average of the given odd numbers from 11 to 951 = 481 Answer
Similar Questions
(1) Find the average of the first 2157 even numbers.
(2) Find the average of even numbers from 6 to 1084
(3) Find the average of the first 4939 even numbers.
(4) Find the average of the first 2550 odd numbers.
(5) Find the average of the first 2857 odd numbers.
(6) Find the average of odd numbers from 11 to 309
(7) Find the average of the first 2508 even numbers.
(8) Find the average of odd numbers from 11 to 249
(9) Find the average of odd numbers from 11 to 1125
(10) Find the average of the first 3013 even numbers.