Question : Find the average of odd numbers from 15 to 1799
Correct Answer 907
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1799
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1799 are
15, 17, 19, . . . . 1799
After observing the above list of the odd numbers from 15 to 1799 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1799 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1799
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1799
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 15 to 1799
= 15 + 1799/2
= 1814/2 = 907
Thus, the average of the odd numbers from 15 to 1799 = 907 Answer
Method (2) to find the average of the odd numbers from 15 to 1799
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1799 are
15, 17, 19, . . . . 1799
The odd numbers from 15 to 1799 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1799
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 15 to 1799
1799 = 15 + (n – 1) × 2
⇒ 1799 = 15 + 2 n – 2
⇒ 1799 = 15 – 2 + 2 n
⇒ 1799 = 13 + 2 n
After transposing 13 to LHS
⇒ 1799 – 13 = 2 n
⇒ 1786 = 2 n
After rearranging the above expression
⇒ 2 n = 1786
After transposing 2 to RHS
⇒ n = 1786/2
⇒ n = 893
Thus, the number of terms of odd numbers from 15 to 1799 = 893
This means 1799 is the 893th term.
Finding the sum of the given odd numbers from 15 to 1799
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 15 to 1799
= 893/2 (15 + 1799)
= 893/2 × 1814
= 893 × 1814/2
= 1619902/2 = 809951
Thus, the sum of all terms of the given odd numbers from 15 to 1799 = 809951
And, the total number of terms = 893
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 15 to 1799
= 809951/893 = 907
Thus, the average of the given odd numbers from 15 to 1799 = 907 Answer
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