Question:
Find the average of odd numbers from 9 to 691
Correct Answer
350
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 691
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 691 are
9, 11, 13, . . . . 691
After observing the above list of the odd numbers from 9 to 691 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 691 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 691
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 691
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 9 to 691
= 9 + 691/2
= 700/2 = 350
Thus, the average of the odd numbers from 9 to 691 = 350 Answer
Method (2) to find the average of the odd numbers from 9 to 691
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 691 are
9, 11, 13, . . . . 691
The odd numbers from 9 to 691 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 691
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 9 to 691
691 = 9 + (n – 1) × 2
⇒ 691 = 9 + 2 n – 2
⇒ 691 = 9 – 2 + 2 n
⇒ 691 = 7 + 2 n
After transposing 7 to LHS
⇒ 691 – 7 = 2 n
⇒ 684 = 2 n
After rearranging the above expression
⇒ 2 n = 684
After transposing 2 to RHS
⇒ n = 684/2
⇒ n = 342
Thus, the number of terms of odd numbers from 9 to 691 = 342
This means 691 is the 342th term.
Finding the sum of the given odd numbers from 9 to 691
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 9 to 691
= 342/2 (9 + 691)
= 342/2 × 700
= 342 × 700/2
= 239400/2 = 119700
Thus, the sum of all terms of the given odd numbers from 9 to 691 = 119700
And, the total number of terms = 342
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 9 to 691
= 119700/342 = 350
Thus, the average of the given odd numbers from 9 to 691 = 350 Answer
Similar Questions
(1) What will be the average of the first 4619 odd numbers?
(2) Find the average of odd numbers from 7 to 345
(3) Find the average of the first 3819 odd numbers.
(4) Find the average of even numbers from 12 to 1942
(5) What is the average of the first 1787 even numbers?
(6) Find the average of odd numbers from 7 to 105
(7) What will be the average of the first 4828 odd numbers?
(8) Find the average of odd numbers from 3 to 1475
(9) Find the average of the first 3492 odd numbers.
(10) Find the average of the first 2539 odd numbers.