Question : Find the average of odd numbers from 15 to 505
Correct Answer 260
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 505
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 505 are
15, 17, 19, . . . . 505
After observing the above list of the odd numbers from 15 to 505 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 505 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 505
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 505
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 505
= 15 + 505/2
= 520/2 = 260
Thus, the average of the odd numbers from 15 to 505 = 260 Answer
Method (2) to find the average of the odd numbers from 15 to 505
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 505 are
15, 17, 19, . . . . 505
The odd numbers from 15 to 505 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 505
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 505
505 = 15 + (n – 1) × 2
⇒ 505 = 15 + 2 n – 2
⇒ 505 = 15 – 2 + 2 n
⇒ 505 = 13 + 2 n
After transposing 13 to LHS
⇒ 505 – 13 = 2 n
⇒ 492 = 2 n
After rearranging the above expression
⇒ 2 n = 492
After transposing 2 to RHS
⇒ n = 492/2
⇒ n = 246
Thus, the number of terms of odd numbers from 15 to 505 = 246
This means 505 is the 246th term.
Finding the sum of the given odd numbers from 15 to 505
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 505
= 246/2 (15 + 505)
= 246/2 × 520
= 246 × 520/2
= 127920/2 = 63960
Thus, the sum of all terms of the given odd numbers from 15 to 505 = 63960
And, the total number of terms = 246
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 505
= 63960/246 = 260
Thus, the average of the given odd numbers from 15 to 505 = 260 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 1111
(2) Find the average of odd numbers from 7 to 521
(3) Find the average of odd numbers from 5 to 643
(4) Find the average of even numbers from 10 to 1296
(5) Find the average of the first 3915 odd numbers.
(6) Find the average of the first 882 odd numbers.
(7) Find the average of even numbers from 8 to 1454
(8) Find the average of the first 2041 odd numbers.
(9) Find the average of the first 1873 odd numbers.
(10) Find the average of even numbers from 6 to 900