Question:
Find the average of odd numbers from 7 to 227
Correct Answer
117
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 227
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 227 are
7, 9, 11, . . . . 227
After observing the above list of the odd numbers from 7 to 227 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 227 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 227
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 227
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 7 to 227
= 7 + 227/2
= 234/2 = 117
Thus, the average of the odd numbers from 7 to 227 = 117 Answer
Method (2) to find the average of the odd numbers from 7 to 227
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 227 are
7, 9, 11, . . . . 227
The odd numbers from 7 to 227 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 227
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 7 to 227
227 = 7 + (n – 1) × 2
⇒ 227 = 7 + 2 n – 2
⇒ 227 = 7 – 2 + 2 n
⇒ 227 = 5 + 2 n
After transposing 5 to LHS
⇒ 227 – 5 = 2 n
⇒ 222 = 2 n
After rearranging the above expression
⇒ 2 n = 222
After transposing 2 to RHS
⇒ n = 222/2
⇒ n = 111
Thus, the number of terms of odd numbers from 7 to 227 = 111
This means 227 is the 111th term.
Finding the sum of the given odd numbers from 7 to 227
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 7 to 227
= 111/2 (7 + 227)
= 111/2 × 234
= 111 × 234/2
= 25974/2 = 12987
Thus, the sum of all terms of the given odd numbers from 7 to 227 = 12987
And, the total number of terms = 111
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 7 to 227
= 12987/111 = 117
Thus, the average of the given odd numbers from 7 to 227 = 117 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 1205
(2) What is the average of the first 1518 even numbers?
(3) Find the average of the first 1569 odd numbers.
(4) What is the average of the first 1457 even numbers?
(5) Find the average of the first 2253 odd numbers.
(6) What is the average of the first 1914 even numbers?
(7) Find the average of even numbers from 4 to 1622
(8) Find the average of the first 2312 odd numbers.
(9) Find the average of the first 1709 odd numbers.
(10) What will be the average of the first 4511 odd numbers?