Question : Find the average of odd numbers from 15 to 1771
Correct Answer 893
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1771
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1771 are
15, 17, 19, . . . . 1771
After observing the above list of the odd numbers from 15 to 1771 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1771 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1771
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1771
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 1771
= 15 + 1771/2
= 1786/2 = 893
Thus, the average of the odd numbers from 15 to 1771 = 893 Answer
Method (2) to find the average of the odd numbers from 15 to 1771
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1771 are
15, 17, 19, . . . . 1771
The odd numbers from 15 to 1771 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1771
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 1771
1771 = 15 + (n – 1) × 2
⇒ 1771 = 15 + 2 n – 2
⇒ 1771 = 15 – 2 + 2 n
⇒ 1771 = 13 + 2 n
After transposing 13 to LHS
⇒ 1771 – 13 = 2 n
⇒ 1758 = 2 n
After rearranging the above expression
⇒ 2 n = 1758
After transposing 2 to RHS
⇒ n = 1758/2
⇒ n = 879
Thus, the number of terms of odd numbers from 15 to 1771 = 879
This means 1771 is the 879th term.
Finding the sum of the given odd numbers from 15 to 1771
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 1771
= 879/2 (15 + 1771)
= 879/2 × 1786
= 879 × 1786/2
= 1569894/2 = 784947
Thus, the sum of all terms of the given odd numbers from 15 to 1771 = 784947
And, the total number of terms = 879
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 1771
= 784947/879 = 893
Thus, the average of the given odd numbers from 15 to 1771 = 893 Answer
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