Question : Find the average of odd numbers from 11 to 931
Correct Answer 471
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 931
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 931 are
11, 13, 15, . . . . 931
After observing the above list of the odd numbers from 11 to 931 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 931 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 931
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 931
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 931
= 11 + 931/2
= 942/2 = 471
Thus, the average of the odd numbers from 11 to 931 = 471 Answer
Method (2) to find the average of the odd numbers from 11 to 931
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 931 are
11, 13, 15, . . . . 931
The odd numbers from 11 to 931 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 931
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 931
931 = 11 + (n – 1) × 2
⇒ 931 = 11 + 2 n – 2
⇒ 931 = 11 – 2 + 2 n
⇒ 931 = 9 + 2 n
After transposing 9 to LHS
⇒ 931 – 9 = 2 n
⇒ 922 = 2 n
After rearranging the above expression
⇒ 2 n = 922
After transposing 2 to RHS
⇒ n = 922/2
⇒ n = 461
Thus, the number of terms of odd numbers from 11 to 931 = 461
This means 931 is the 461th term.
Finding the sum of the given odd numbers from 11 to 931
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 931
= 461/2 (11 + 931)
= 461/2 × 942
= 461 × 942/2
= 434262/2 = 217131
Thus, the sum of all terms of the given odd numbers from 11 to 931 = 217131
And, the total number of terms = 461
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 931
= 217131/461 = 471
Thus, the average of the given odd numbers from 11 to 931 = 471 Answer
Similar Questions
(1) Find the average of odd numbers from 11 to 867
(2) Find the average of even numbers from 12 to 758
(3) Find the average of even numbers from 12 to 258
(4) Find the average of even numbers from 8 to 1226
(5) What will be the average of the first 4035 odd numbers?
(6) Find the average of even numbers from 10 to 490
(7) Find the average of odd numbers from 3 to 1029
(8) What is the average of the first 65 even numbers?
(9) Find the average of even numbers from 12 to 336
(10) Find the average of the first 3286 odd numbers.