Question : Find the average of odd numbers from 3 to 889
Correct Answer 446
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 889
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 889 are
3, 5, 7, . . . . 889
After observing the above list of the odd numbers from 3 to 889 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 889 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 889
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 889
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 889
= 3 + 889/2
= 892/2 = 446
Thus, the average of the odd numbers from 3 to 889 = 446 Answer
Method (2) to find the average of the odd numbers from 3 to 889
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 889 are
3, 5, 7, . . . . 889
The odd numbers from 3 to 889 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 889
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 889
889 = 3 + (n – 1) × 2
⇒ 889 = 3 + 2 n – 2
⇒ 889 = 3 – 2 + 2 n
⇒ 889 = 1 + 2 n
After transposing 1 to LHS
⇒ 889 – 1 = 2 n
⇒ 888 = 2 n
After rearranging the above expression
⇒ 2 n = 888
After transposing 2 to RHS
⇒ n = 888/2
⇒ n = 444
Thus, the number of terms of odd numbers from 3 to 889 = 444
This means 889 is the 444th term.
Finding the sum of the given odd numbers from 3 to 889
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 889
= 444/2 (3 + 889)
= 444/2 × 892
= 444 × 892/2
= 396048/2 = 198024
Thus, the sum of all terms of the given odd numbers from 3 to 889 = 198024
And, the total number of terms = 444
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 889
= 198024/444 = 446
Thus, the average of the given odd numbers from 3 to 889 = 446 Answer
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