Question:
Find the average of odd numbers from 15 to 1663
Correct Answer
839
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1663
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1663 are
15, 17, 19, . . . . 1663
After observing the above list of the odd numbers from 15 to 1663 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1663 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1663
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1663
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 1663
= 15 + 1663/2
= 1678/2 = 839
Thus, the average of the odd numbers from 15 to 1663 = 839 Answer
Method (2) to find the average of the odd numbers from 15 to 1663
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1663 are
15, 17, 19, . . . . 1663
The odd numbers from 15 to 1663 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1663
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 1663
1663 = 15 + (n – 1) × 2
⇒ 1663 = 15 + 2 n – 2
⇒ 1663 = 15 – 2 + 2 n
⇒ 1663 = 13 + 2 n
After transposing 13 to LHS
⇒ 1663 – 13 = 2 n
⇒ 1650 = 2 n
After rearranging the above expression
⇒ 2 n = 1650
After transposing 2 to RHS
⇒ n = 1650/2
⇒ n = 825
Thus, the number of terms of odd numbers from 15 to 1663 = 825
This means 1663 is the 825th term.
Finding the sum of the given odd numbers from 15 to 1663
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 1663
= 825/2 (15 + 1663)
= 825/2 × 1678
= 825 × 1678/2
= 1384350/2 = 692175
Thus, the sum of all terms of the given odd numbers from 15 to 1663 = 692175
And, the total number of terms = 825
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 1663
= 692175/825 = 839
Thus, the average of the given odd numbers from 15 to 1663 = 839 Answer
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