Question:
Find the average of odd numbers from 11 to 1151
Correct Answer
581
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 1151
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 1151 are
11, 13, 15, . . . . 1151
After observing the above list of the odd numbers from 11 to 1151 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 1151 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 1151
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1151
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 11 to 1151
= 11 + 1151/2
= 1162/2 = 581
Thus, the average of the odd numbers from 11 to 1151 = 581 Answer
Method (2) to find the average of the odd numbers from 11 to 1151
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 1151 are
11, 13, 15, . . . . 1151
The odd numbers from 11 to 1151 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1151
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 11 to 1151
1151 = 11 + (n – 1) × 2
⇒ 1151 = 11 + 2 n – 2
⇒ 1151 = 11 – 2 + 2 n
⇒ 1151 = 9 + 2 n
After transposing 9 to LHS
⇒ 1151 – 9 = 2 n
⇒ 1142 = 2 n
After rearranging the above expression
⇒ 2 n = 1142
After transposing 2 to RHS
⇒ n = 1142/2
⇒ n = 571
Thus, the number of terms of odd numbers from 11 to 1151 = 571
This means 1151 is the 571th term.
Finding the sum of the given odd numbers from 11 to 1151
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 11 to 1151
= 571/2 (11 + 1151)
= 571/2 × 1162
= 571 × 1162/2
= 663502/2 = 331751
Thus, the sum of all terms of the given odd numbers from 11 to 1151 = 331751
And, the total number of terms = 571
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 11 to 1151
= 331751/571 = 581
Thus, the average of the given odd numbers from 11 to 1151 = 581 Answer
Similar Questions
(1) Find the average of the first 266 odd numbers.
(2) Find the average of the first 1290 odd numbers.
(3) Find the average of even numbers from 4 to 1674
(4) Find the average of the first 1276 odd numbers.
(5) Find the average of even numbers from 4 to 1436
(6) Find the average of odd numbers from 13 to 465
(7) Find the average of even numbers from 6 to 126
(8) Find the average of even numbers from 12 to 780
(9) Find the average of odd numbers from 3 to 1247
(10) Find the average of even numbers from 12 to 262