Question:
Find the average of odd numbers from 3 to 861
Correct Answer
432
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 861
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 861 are
3, 5, 7, . . . . 861
After observing the above list of the odd numbers from 3 to 861 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 861 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 861
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 861
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 3 to 861
= 3 + 861/2
= 864/2 = 432
Thus, the average of the odd numbers from 3 to 861 = 432 Answer
Method (2) to find the average of the odd numbers from 3 to 861
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 861 are
3, 5, 7, . . . . 861
The odd numbers from 3 to 861 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 861
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 3 to 861
861 = 3 + (n – 1) × 2
⇒ 861 = 3 + 2 n – 2
⇒ 861 = 3 – 2 + 2 n
⇒ 861 = 1 + 2 n
After transposing 1 to LHS
⇒ 861 – 1 = 2 n
⇒ 860 = 2 n
After rearranging the above expression
⇒ 2 n = 860
After transposing 2 to RHS
⇒ n = 860/2
⇒ n = 430
Thus, the number of terms of odd numbers from 3 to 861 = 430
This means 861 is the 430th term.
Finding the sum of the given odd numbers from 3 to 861
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 3 to 861
= 430/2 (3 + 861)
= 430/2 × 864
= 430 × 864/2
= 371520/2 = 185760
Thus, the sum of all terms of the given odd numbers from 3 to 861 = 185760
And, the total number of terms = 430
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 3 to 861
= 185760/430 = 432
Thus, the average of the given odd numbers from 3 to 861 = 432 Answer
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