Question:
Find the average of odd numbers from 9 to 1113
Correct Answer
561
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 1113
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 1113 are
9, 11, 13, . . . . 1113
After observing the above list of the odd numbers from 9 to 1113 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 1113 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 1113
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 1113
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 1113
= 9 + 1113/2
= 1122/2 = 561
Thus, the average of the odd numbers from 9 to 1113 = 561 Answer
Method (2) to find the average of the odd numbers from 9 to 1113
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 1113 are
9, 11, 13, . . . . 1113
The odd numbers from 9 to 1113 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 1113
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 1113
1113 = 9 + (n – 1) × 2
⇒ 1113 = 9 + 2 n – 2
⇒ 1113 = 9 – 2 + 2 n
⇒ 1113 = 7 + 2 n
After transposing 7 to LHS
⇒ 1113 – 7 = 2 n
⇒ 1106 = 2 n
After rearranging the above expression
⇒ 2 n = 1106
After transposing 2 to RHS
⇒ n = 1106/2
⇒ n = 553
Thus, the number of terms of odd numbers from 9 to 1113 = 553
This means 1113 is the 553th term.
Finding the sum of the given odd numbers from 9 to 1113
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 1113
= 553/2 (9 + 1113)
= 553/2 × 1122
= 553 × 1122/2
= 620466/2 = 310233
Thus, the sum of all terms of the given odd numbers from 9 to 1113 = 310233
And, the total number of terms = 553
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 1113
= 310233/553 = 561
Thus, the average of the given odd numbers from 9 to 1113 = 561 Answer
Similar Questions
(1) What is the average of the first 694 even numbers?
(2) Find the average of the first 3111 even numbers.
(3) Find the average of the first 3482 odd numbers.
(4) What is the average of the first 1023 even numbers?
(5) Find the average of odd numbers from 3 to 511
(6) Find the average of odd numbers from 3 to 1109
(7) What is the average of the first 1308 even numbers?
(8) Find the average of even numbers from 10 to 1752
(9) Find the average of odd numbers from 11 to 593
(10) Find the average of odd numbers from 5 to 1093