Question:
Find the average of odd numbers from 7 to 243
Correct Answer
125
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 243
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 243 are
7, 9, 11, . . . . 243
After observing the above list of the odd numbers from 7 to 243 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 243 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 243
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 243
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 243
= 7 + 243/2
= 250/2 = 125
Thus, the average of the odd numbers from 7 to 243 = 125 Answer
Method (2) to find the average of the odd numbers from 7 to 243
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 243 are
7, 9, 11, . . . . 243
The odd numbers from 7 to 243 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 243
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 243
243 = 7 + (n – 1) × 2
⇒ 243 = 7 + 2 n – 2
⇒ 243 = 7 – 2 + 2 n
⇒ 243 = 5 + 2 n
After transposing 5 to LHS
⇒ 243 – 5 = 2 n
⇒ 238 = 2 n
After rearranging the above expression
⇒ 2 n = 238
After transposing 2 to RHS
⇒ n = 238/2
⇒ n = 119
Thus, the number of terms of odd numbers from 7 to 243 = 119
This means 243 is the 119th term.
Finding the sum of the given odd numbers from 7 to 243
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 243
= 119/2 (7 + 243)
= 119/2 × 250
= 119 × 250/2
= 29750/2 = 14875
Thus, the sum of all terms of the given odd numbers from 7 to 243 = 14875
And, the total number of terms = 119
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 243
= 14875/119 = 125
Thus, the average of the given odd numbers from 7 to 243 = 125 Answer
Similar Questions
(1) Find the average of odd numbers from 5 to 1021
(2) Find the average of odd numbers from 9 to 371
(3) Find the average of odd numbers from 11 to 1031
(4) What is the average of the first 195 odd numbers?
(5) Find the average of odd numbers from 15 to 377
(6) Find the average of odd numbers from 15 to 607
(7) What is the average of the first 1169 even numbers?
(8) Find the average of the first 4377 even numbers.
(9) Find the average of odd numbers from 5 to 749
(10) Find the average of even numbers from 10 to 168