Question:
Find the average of odd numbers from 5 to 675
Correct Answer
340
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 675
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 675 are
5, 7, 9, . . . . 675
After observing the above list of the odd numbers from 5 to 675 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 675 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 675
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 675
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 5 to 675
= 5 + 675/2
= 680/2 = 340
Thus, the average of the odd numbers from 5 to 675 = 340 Answer
Method (2) to find the average of the odd numbers from 5 to 675
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 675 are
5, 7, 9, . . . . 675
The odd numbers from 5 to 675 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 675
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 5 to 675
675 = 5 + (n – 1) × 2
⇒ 675 = 5 + 2 n – 2
⇒ 675 = 5 – 2 + 2 n
⇒ 675 = 3 + 2 n
After transposing 3 to LHS
⇒ 675 – 3 = 2 n
⇒ 672 = 2 n
After rearranging the above expression
⇒ 2 n = 672
After transposing 2 to RHS
⇒ n = 672/2
⇒ n = 336
Thus, the number of terms of odd numbers from 5 to 675 = 336
This means 675 is the 336th term.
Finding the sum of the given odd numbers from 5 to 675
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 5 to 675
= 336/2 (5 + 675)
= 336/2 × 680
= 336 × 680/2
= 228480/2 = 114240
Thus, the sum of all terms of the given odd numbers from 5 to 675 = 114240
And, the total number of terms = 336
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 5 to 675
= 114240/336 = 340
Thus, the average of the given odd numbers from 5 to 675 = 340 Answer
Similar Questions
(1) Find the average of the first 2831 even numbers.
(2) Find the average of even numbers from 4 to 358
(3) What will be the average of the first 4710 odd numbers?
(4) Find the average of the first 3028 even numbers.
(5) Find the average of odd numbers from 11 to 1079
(6) Find the average of even numbers from 10 to 404
(7) What will be the average of the first 4055 odd numbers?
(8) What is the average of the first 671 even numbers?
(9) Find the average of even numbers from 12 to 1926
(10) Find the average of even numbers from 6 to 410