Question:
Find the average of odd numbers from 11 to 1125
Correct Answer
568
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 1125
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 1125 are
11, 13, 15, . . . . 1125
After observing the above list of the odd numbers from 11 to 1125 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 1125 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 1125
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1125
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 1125
= 11 + 1125/2
= 1136/2 = 568
Thus, the average of the odd numbers from 11 to 1125 = 568 Answer
Method (2) to find the average of the odd numbers from 11 to 1125
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 1125 are
11, 13, 15, . . . . 1125
The odd numbers from 11 to 1125 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1125
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 1125
1125 = 11 + (n – 1) × 2
⇒ 1125 = 11 + 2 n – 2
⇒ 1125 = 11 – 2 + 2 n
⇒ 1125 = 9 + 2 n
After transposing 9 to LHS
⇒ 1125 – 9 = 2 n
⇒ 1116 = 2 n
After rearranging the above expression
⇒ 2 n = 1116
After transposing 2 to RHS
⇒ n = 1116/2
⇒ n = 558
Thus, the number of terms of odd numbers from 11 to 1125 = 558
This means 1125 is the 558th term.
Finding the sum of the given odd numbers from 11 to 1125
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 1125
= 558/2 (11 + 1125)
= 558/2 × 1136
= 558 × 1136/2
= 633888/2 = 316944
Thus, the sum of all terms of the given odd numbers from 11 to 1125 = 316944
And, the total number of terms = 558
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 1125
= 316944/558 = 568
Thus, the average of the given odd numbers from 11 to 1125 = 568 Answer
Similar Questions
(1) Find the average of the first 656 odd numbers.
(2) Find the average of odd numbers from 13 to 219
(3) Find the average of odd numbers from 7 to 1315
(4) Find the average of odd numbers from 11 to 67
(5) What will be the average of the first 4330 odd numbers?
(6) Find the average of odd numbers from 13 to 789
(7) Find the average of the first 4552 even numbers.
(8) Find the average of the first 2385 even numbers.
(9) What will be the average of the first 4057 odd numbers?
(10) Find the average of the first 591 odd numbers.