Question : Find the average of odd numbers from 7 to 979
Correct Answer 493
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 979
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 979 are
7, 9, 11, . . . . 979
After observing the above list of the odd numbers from 7 to 979 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 979 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 979
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 979
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 979
= 7 + 979/2
= 986/2 = 493
Thus, the average of the odd numbers from 7 to 979 = 493 Answer
Method (2) to find the average of the odd numbers from 7 to 979
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 979 are
7, 9, 11, . . . . 979
The odd numbers from 7 to 979 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 979
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 979
979 = 7 + (n – 1) × 2
⇒ 979 = 7 + 2 n – 2
⇒ 979 = 7 – 2 + 2 n
⇒ 979 = 5 + 2 n
After transposing 5 to LHS
⇒ 979 – 5 = 2 n
⇒ 974 = 2 n
After rearranging the above expression
⇒ 2 n = 974
After transposing 2 to RHS
⇒ n = 974/2
⇒ n = 487
Thus, the number of terms of odd numbers from 7 to 979 = 487
This means 979 is the 487th term.
Finding the sum of the given odd numbers from 7 to 979
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 979
= 487/2 (7 + 979)
= 487/2 × 986
= 487 × 986/2
= 480182/2 = 240091
Thus, the sum of all terms of the given odd numbers from 7 to 979 = 240091
And, the total number of terms = 487
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 979
= 240091/487 = 493
Thus, the average of the given odd numbers from 7 to 979 = 493 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 921
(2) Find the average of even numbers from 10 to 1156
(3) Find the average of odd numbers from 15 to 1447
(4) Find the average of odd numbers from 9 to 1369
(5) Find the average of even numbers from 4 to 640
(6) Find the average of odd numbers from 5 to 1005
(7) What is the average of the first 368 even numbers?
(8) Find the average of the first 3275 odd numbers.
(9) Find the average of the first 1809 odd numbers.
(10) Find the average of odd numbers from 9 to 573