Question : Find the average of odd numbers from 11 to 1261
Correct Answer 636
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 1261
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 1261 are
11, 13, 15, . . . . 1261
After observing the above list of the odd numbers from 11 to 1261 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 1261 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 1261
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1261
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 1261
= 11 + 1261/2
= 1272/2 = 636
Thus, the average of the odd numbers from 11 to 1261 = 636 Answer
Method (2) to find the average of the odd numbers from 11 to 1261
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 1261 are
11, 13, 15, . . . . 1261
The odd numbers from 11 to 1261 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1261
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 1261
1261 = 11 + (n – 1) × 2
⇒ 1261 = 11 + 2 n – 2
⇒ 1261 = 11 – 2 + 2 n
⇒ 1261 = 9 + 2 n
After transposing 9 to LHS
⇒ 1261 – 9 = 2 n
⇒ 1252 = 2 n
After rearranging the above expression
⇒ 2 n = 1252
After transposing 2 to RHS
⇒ n = 1252/2
⇒ n = 626
Thus, the number of terms of odd numbers from 11 to 1261 = 626
This means 1261 is the 626th term.
Finding the sum of the given odd numbers from 11 to 1261
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 1261
= 626/2 (11 + 1261)
= 626/2 × 1272
= 626 × 1272/2
= 796272/2 = 398136
Thus, the sum of all terms of the given odd numbers from 11 to 1261 = 398136
And, the total number of terms = 626
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 1261
= 398136/626 = 636
Thus, the average of the given odd numbers from 11 to 1261 = 636 Answer
Similar Questions
(1) What is the average of the first 648 even numbers?
(2) Find the average of the first 2141 even numbers.
(3) Find the average of odd numbers from 7 to 1343
(4) Find the average of odd numbers from 9 to 281
(5) Find the average of even numbers from 10 to 116
(6) What is the average of the first 1861 even numbers?
(7) Find the average of even numbers from 6 to 1294
(8) Find the average of the first 4686 even numbers.
(9) What will be the average of the first 4741 odd numbers?
(10) What is the average of the first 1628 even numbers?