Question:
Find the average of odd numbers from 9 to 1401
Correct Answer
705
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 1401
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 1401 are
9, 11, 13, . . . . 1401
After observing the above list of the odd numbers from 9 to 1401 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 1401 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 1401
The First Term (a) = 9
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 9 to 1401
= 9 + 1401/2
= 1410/2 = 705
Thus, the average of the odd numbers from 9 to 1401 = 705 Answer
Method (2) to find the average of the odd numbers from 9 to 1401
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 1401 are
9, 11, 13, . . . . 1401
The odd numbers from 9 to 1401 form an Arithmetic Series in which
The First Term (a) = 9
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 9 to 1401
1401 = 9 + (n – 1) × 2
⇒ 1401 = 9 + 2 n – 2
⇒ 1401 = 9 – 2 + 2 n
⇒ 1401 = 7 + 2 n
After transposing 7 to LHS
⇒ 1401 – 7 = 2 n
⇒ 1394 = 2 n
After rearranging the above expression
⇒ 2 n = 1394
After transposing 2 to RHS
⇒ n = 1394/2
⇒ n = 697
Thus, the number of terms of odd numbers from 9 to 1401 = 697
This means 1401 is the 697th term.
Finding the sum of the given odd numbers from 9 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 9 to 1401
= 697/2 (9 + 1401)
= 697/2 × 1410
= 697 × 1410/2
= 982770/2 = 491385
Thus, the sum of all terms of the given odd numbers from 9 to 1401 = 491385
And, the total number of terms = 697
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 1401
= 491385/697 = 705
Thus, the average of the given odd numbers from 9 to 1401 = 705 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 1449
(2) Find the average of odd numbers from 11 to 1231
(3) Find the average of odd numbers from 11 to 377
(4) Find the average of odd numbers from 7 to 657
(5) Find the average of the first 2861 odd numbers.
(6) Find the average of the first 4441 even numbers.
(7) Find the average of even numbers from 12 to 1572
(8) Find the average of odd numbers from 13 to 1195
(9) Find the average of the first 1536 odd numbers.
(10) Find the average of even numbers from 10 to 972