Question : Find the average of odd numbers from 7 to 601
Correct Answer 304
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 601
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 601 are
7, 9, 11, . . . . 601
After observing the above list of the odd numbers from 7 to 601 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 601 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 601
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 601
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 601
= 7 + 601/2
= 608/2 = 304
Thus, the average of the odd numbers from 7 to 601 = 304 Answer
Method (2) to find the average of the odd numbers from 7 to 601
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 601 are
7, 9, 11, . . . . 601
The odd numbers from 7 to 601 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 601
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 601
601 = 7 + (n – 1) × 2
⇒ 601 = 7 + 2 n – 2
⇒ 601 = 7 – 2 + 2 n
⇒ 601 = 5 + 2 n
After transposing 5 to LHS
⇒ 601 – 5 = 2 n
⇒ 596 = 2 n
After rearranging the above expression
⇒ 2 n = 596
After transposing 2 to RHS
⇒ n = 596/2
⇒ n = 298
Thus, the number of terms of odd numbers from 7 to 601 = 298
This means 601 is the 298th term.
Finding the sum of the given odd numbers from 7 to 601
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 601
= 298/2 (7 + 601)
= 298/2 × 608
= 298 × 608/2
= 181184/2 = 90592
Thus, the sum of all terms of the given odd numbers from 7 to 601 = 90592
And, the total number of terms = 298
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 601
= 90592/298 = 304
Thus, the average of the given odd numbers from 7 to 601 = 304 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 1220
(2) Find the average of odd numbers from 15 to 507
(3) What is the average of the first 1425 even numbers?
(4) Find the average of the first 4726 even numbers.
(5) What is the average of the first 1479 even numbers?
(6) Find the average of the first 2209 odd numbers.
(7) Find the average of odd numbers from 5 to 473
(8) Find the average of odd numbers from 11 to 1177
(9) Find the average of the first 1689 odd numbers.
(10) Find the average of even numbers from 10 to 406