Question : Find the average of odd numbers from 13 to 863
Correct Answer 438
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 863
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 863 are
13, 15, 17, . . . . 863
After observing the above list of the odd numbers from 13 to 863 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 863 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 863
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 863
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 13 to 863
= 13 + 863/2
= 876/2 = 438
Thus, the average of the odd numbers from 13 to 863 = 438 Answer
Method (2) to find the average of the odd numbers from 13 to 863
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 863 are
13, 15, 17, . . . . 863
The odd numbers from 13 to 863 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 863
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 13 to 863
863 = 13 + (n – 1) × 2
⇒ 863 = 13 + 2 n – 2
⇒ 863 = 13 – 2 + 2 n
⇒ 863 = 11 + 2 n
After transposing 11 to LHS
⇒ 863 – 11 = 2 n
⇒ 852 = 2 n
After rearranging the above expression
⇒ 2 n = 852
After transposing 2 to RHS
⇒ n = 852/2
⇒ n = 426
Thus, the number of terms of odd numbers from 13 to 863 = 426
This means 863 is the 426th term.
Finding the sum of the given odd numbers from 13 to 863
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 13 to 863
= 426/2 (13 + 863)
= 426/2 × 876
= 426 × 876/2
= 373176/2 = 186588
Thus, the sum of all terms of the given odd numbers from 13 to 863 = 186588
And, the total number of terms = 426
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 13 to 863
= 186588/426 = 438
Thus, the average of the given odd numbers from 13 to 863 = 438 Answer
Similar Questions
(1) What will be the average of the first 4257 odd numbers?
(2) Find the average of the first 3829 even numbers.
(3) Find the average of the first 3199 odd numbers.
(4) Find the average of the first 2489 even numbers.
(5) Find the average of even numbers from 12 to 1900
(6) Find the average of the first 3676 even numbers.
(7) Find the average of the first 743 odd numbers.
(8) Find the average of odd numbers from 3 to 1143
(9) Find the average of the first 3563 even numbers.
(10) Find the average of the first 2940 even numbers.