Question : Find the average of odd numbers from 11 to 541
Correct Answer 276
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 541
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 541 are
11, 13, 15, . . . . 541
After observing the above list of the odd numbers from 11 to 541 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 541 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 541
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 541
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 541
= 11 + 541/2
= 552/2 = 276
Thus, the average of the odd numbers from 11 to 541 = 276 Answer
Method (2) to find the average of the odd numbers from 11 to 541
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 541 are
11, 13, 15, . . . . 541
The odd numbers from 11 to 541 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 541
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 541
541 = 11 + (n – 1) × 2
⇒ 541 = 11 + 2 n – 2
⇒ 541 = 11 – 2 + 2 n
⇒ 541 = 9 + 2 n
After transposing 9 to LHS
⇒ 541 – 9 = 2 n
⇒ 532 = 2 n
After rearranging the above expression
⇒ 2 n = 532
After transposing 2 to RHS
⇒ n = 532/2
⇒ n = 266
Thus, the number of terms of odd numbers from 11 to 541 = 266
This means 541 is the 266th term.
Finding the sum of the given odd numbers from 11 to 541
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 541
= 266/2 (11 + 541)
= 266/2 × 552
= 266 × 552/2
= 146832/2 = 73416
Thus, the sum of all terms of the given odd numbers from 11 to 541 = 73416
And, the total number of terms = 266
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 541
= 73416/266 = 276
Thus, the average of the given odd numbers from 11 to 541 = 276 Answer
Similar Questions
(1) Find the average of the first 4866 even numbers.
(2) Find the average of the first 259 odd numbers.
(3) What will be the average of the first 4174 odd numbers?
(4) Find the average of the first 4714 even numbers.
(5) Find the average of even numbers from 12 to 1868
(6) Find the average of odd numbers from 15 to 757
(7) Find the average of even numbers from 12 to 1240
(8) Find the average of odd numbers from 11 to 501
(9) Find the average of the first 4918 even numbers.
(10) Find the average of even numbers from 10 to 1378