Question:
Find the average of odd numbers from 5 to 693
Correct Answer
349
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 693
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 693 are
5, 7, 9, . . . . 693
After observing the above list of the odd numbers from 5 to 693 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 693 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 693
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 693
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 5 to 693
= 5 + 693/2
= 698/2 = 349
Thus, the average of the odd numbers from 5 to 693 = 349 Answer
Method (2) to find the average of the odd numbers from 5 to 693
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 693 are
5, 7, 9, . . . . 693
The odd numbers from 5 to 693 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 693
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 5 to 693
693 = 5 + (n – 1) × 2
⇒ 693 = 5 + 2 n – 2
⇒ 693 = 5 – 2 + 2 n
⇒ 693 = 3 + 2 n
After transposing 3 to LHS
⇒ 693 – 3 = 2 n
⇒ 690 = 2 n
After rearranging the above expression
⇒ 2 n = 690
After transposing 2 to RHS
⇒ n = 690/2
⇒ n = 345
Thus, the number of terms of odd numbers from 5 to 693 = 345
This means 693 is the 345th term.
Finding the sum of the given odd numbers from 5 to 693
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 5 to 693
= 345/2 (5 + 693)
= 345/2 × 698
= 345 × 698/2
= 240810/2 = 120405
Thus, the sum of all terms of the given odd numbers from 5 to 693 = 120405
And, the total number of terms = 345
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 5 to 693
= 120405/345 = 349
Thus, the average of the given odd numbers from 5 to 693 = 349 Answer
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