Question:
Find the average of odd numbers from 9 to 249
Correct Answer
129
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 249
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 249 are
9, 11, 13, . . . . 249
After observing the above list of the odd numbers from 9 to 249 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 249 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 249
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 249
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 9 to 249
= 9 + 249/2
= 258/2 = 129
Thus, the average of the odd numbers from 9 to 249 = 129 Answer
Method (2) to find the average of the odd numbers from 9 to 249
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 249 are
9, 11, 13, . . . . 249
The odd numbers from 9 to 249 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 249
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 9 to 249
249 = 9 + (n – 1) × 2
⇒ 249 = 9 + 2 n – 2
⇒ 249 = 9 – 2 + 2 n
⇒ 249 = 7 + 2 n
After transposing 7 to LHS
⇒ 249 – 7 = 2 n
⇒ 242 = 2 n
After rearranging the above expression
⇒ 2 n = 242
After transposing 2 to RHS
⇒ n = 242/2
⇒ n = 121
Thus, the number of terms of odd numbers from 9 to 249 = 121
This means 249 is the 121th term.
Finding the sum of the given odd numbers from 9 to 249
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 9 to 249
= 121/2 (9 + 249)
= 121/2 × 258
= 121 × 258/2
= 31218/2 = 15609
Thus, the sum of all terms of the given odd numbers from 9 to 249 = 15609
And, the total number of terms = 121
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 9 to 249
= 15609/121 = 129
Thus, the average of the given odd numbers from 9 to 249 = 129 Answer
Similar Questions
(1) Find the average of odd numbers from 11 to 625
(2) Find the average of the first 2499 odd numbers.
(3) Find the average of odd numbers from 9 to 427
(4) Find the average of the first 2559 odd numbers.
(5) What will be the average of the first 4437 odd numbers?
(6) Find the average of even numbers from 4 to 2000
(7) What will be the average of the first 4630 odd numbers?
(8) Find the average of even numbers from 10 to 1010
(9) Find the average of even numbers from 6 to 392
(10) Find the average of odd numbers from 7 to 991