Question:
Find the average of odd numbers from 3 to 1097
Correct Answer
550
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 1097
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 1097 are
3, 5, 7, . . . . 1097
After observing the above list of the odd numbers from 3 to 1097 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 1097 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 1097
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1097
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 3 to 1097
= 3 + 1097/2
= 1100/2 = 550
Thus, the average of the odd numbers from 3 to 1097 = 550 Answer
Method (2) to find the average of the odd numbers from 3 to 1097
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 1097 are
3, 5, 7, . . . . 1097
The odd numbers from 3 to 1097 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1097
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 3 to 1097
1097 = 3 + (n – 1) × 2
⇒ 1097 = 3 + 2 n – 2
⇒ 1097 = 3 – 2 + 2 n
⇒ 1097 = 1 + 2 n
After transposing 1 to LHS
⇒ 1097 – 1 = 2 n
⇒ 1096 = 2 n
After rearranging the above expression
⇒ 2 n = 1096
After transposing 2 to RHS
⇒ n = 1096/2
⇒ n = 548
Thus, the number of terms of odd numbers from 3 to 1097 = 548
This means 1097 is the 548th term.
Finding the sum of the given odd numbers from 3 to 1097
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 3 to 1097
= 548/2 (3 + 1097)
= 548/2 × 1100
= 548 × 1100/2
= 602800/2 = 301400
Thus, the sum of all terms of the given odd numbers from 3 to 1097 = 301400
And, the total number of terms = 548
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 3 to 1097
= 301400/548 = 550
Thus, the average of the given odd numbers from 3 to 1097 = 550 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 916
(2) Find the average of the first 1771 odd numbers.
(3) Find the average of the first 2589 odd numbers.
(4) Find the average of the first 642 odd numbers.
(5) Find the average of even numbers from 4 to 1328
(6) What will be the average of the first 4212 odd numbers?
(7) Find the average of even numbers from 12 to 982
(8) Find the average of the first 1941 odd numbers.
(9) Find the average of odd numbers from 5 to 1015
(10) Find the average of the first 4435 even numbers.