Question : Find the average of odd numbers from 15 to 1653
Correct Answer 834
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1653
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1653 are
15, 17, 19, . . . . 1653
After observing the above list of the odd numbers from 15 to 1653 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1653 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1653
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1653
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 1653
= 15 + 1653/2
= 1668/2 = 834
Thus, the average of the odd numbers from 15 to 1653 = 834 Answer
Method (2) to find the average of the odd numbers from 15 to 1653
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1653 are
15, 17, 19, . . . . 1653
The odd numbers from 15 to 1653 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1653
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 1653
1653 = 15 + (n – 1) × 2
⇒ 1653 = 15 + 2 n – 2
⇒ 1653 = 15 – 2 + 2 n
⇒ 1653 = 13 + 2 n
After transposing 13 to LHS
⇒ 1653 – 13 = 2 n
⇒ 1640 = 2 n
After rearranging the above expression
⇒ 2 n = 1640
After transposing 2 to RHS
⇒ n = 1640/2
⇒ n = 820
Thus, the number of terms of odd numbers from 15 to 1653 = 820
This means 1653 is the 820th term.
Finding the sum of the given odd numbers from 15 to 1653
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 1653
= 820/2 (15 + 1653)
= 820/2 × 1668
= 820 × 1668/2
= 1367760/2 = 683880
Thus, the sum of all terms of the given odd numbers from 15 to 1653 = 683880
And, the total number of terms = 820
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 1653
= 683880/820 = 834
Thus, the average of the given odd numbers from 15 to 1653 = 834 Answer
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