Question : Find the average of odd numbers from 15 to 489
Correct Answer 252
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 489
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 489 are
15, 17, 19, . . . . 489
After observing the above list of the odd numbers from 15 to 489 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 489 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 489
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 489
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 15 to 489
= 15 + 489/2
= 504/2 = 252
Thus, the average of the odd numbers from 15 to 489 = 252 Answer
Method (2) to find the average of the odd numbers from 15 to 489
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 489 are
15, 17, 19, . . . . 489
The odd numbers from 15 to 489 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 489
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 15 to 489
489 = 15 + (n – 1) × 2
⇒ 489 = 15 + 2 n – 2
⇒ 489 = 15 – 2 + 2 n
⇒ 489 = 13 + 2 n
After transposing 13 to LHS
⇒ 489 – 13 = 2 n
⇒ 476 = 2 n
After rearranging the above expression
⇒ 2 n = 476
After transposing 2 to RHS
⇒ n = 476/2
⇒ n = 238
Thus, the number of terms of odd numbers from 15 to 489 = 238
This means 489 is the 238th term.
Finding the sum of the given odd numbers from 15 to 489
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 15 to 489
= 238/2 (15 + 489)
= 238/2 × 504
= 238 × 504/2
= 119952/2 = 59976
Thus, the sum of all terms of the given odd numbers from 15 to 489 = 59976
And, the total number of terms = 238
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 15 to 489
= 59976/238 = 252
Thus, the average of the given odd numbers from 15 to 489 = 252 Answer
Similar Questions
(1) Find the average of the first 376 odd numbers.
(2) What will be the average of the first 4799 odd numbers?
(3) What will be the average of the first 4532 odd numbers?
(4) Find the average of odd numbers from 7 to 663
(5) What is the average of the first 418 even numbers?
(6) Find the average of even numbers from 12 to 216
(7) Find the average of even numbers from 12 to 1666
(8) Find the average of the first 4168 even numbers.
(9) Find the average of odd numbers from 3 to 221
(10) Find the average of the first 4697 even numbers.