Question : Find the average of odd numbers from 11 to 1401
Correct Answer 706
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 1401
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 1401 are
11, 13, 15, . . . . 1401
After observing the above list of the odd numbers from 11 to 1401 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 1401 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 1401
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1401
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 1401
= 11 + 1401/2
= 1412/2 = 706
Thus, the average of the odd numbers from 11 to 1401 = 706 Answer
Method (2) to find the average of the odd numbers from 11 to 1401
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 1401 are
11, 13, 15, . . . . 1401
The odd numbers from 11 to 1401 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1401
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 1401
1401 = 11 + (n – 1) × 2
⇒ 1401 = 11 + 2 n – 2
⇒ 1401 = 11 – 2 + 2 n
⇒ 1401 = 9 + 2 n
After transposing 9 to LHS
⇒ 1401 – 9 = 2 n
⇒ 1392 = 2 n
After rearranging the above expression
⇒ 2 n = 1392
After transposing 2 to RHS
⇒ n = 1392/2
⇒ n = 696
Thus, the number of terms of odd numbers from 11 to 1401 = 696
This means 1401 is the 696th term.
Finding the sum of the given odd numbers from 11 to 1401
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 1401
= 696/2 (11 + 1401)
= 696/2 × 1412
= 696 × 1412/2
= 982752/2 = 491376
Thus, the sum of all terms of the given odd numbers from 11 to 1401 = 491376
And, the total number of terms = 696
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 1401
= 491376/696 = 706
Thus, the average of the given odd numbers from 11 to 1401 = 706 Answer
Similar Questions
(1) What is the average of the first 1445 even numbers?
(2) Find the average of odd numbers from 5 to 1233
(3) Find the average of even numbers from 4 to 1836
(5) Find the average of even numbers from 8 to 1164
(6) Find the average of even numbers from 10 to 242
(7) What will be the average of the first 4933 odd numbers?
(8) What will be the average of the first 4919 odd numbers?
(9) Find the average of the first 3724 even numbers.
(10) What is the average of the first 1262 even numbers?