Question:
Find the average of odd numbers from 13 to 835
Correct Answer
424
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 835
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 835 are
13, 15, 17, . . . . 835
After observing the above list of the odd numbers from 13 to 835 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 835 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 835
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 835
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 835
= 13 + 835/2
= 848/2 = 424
Thus, the average of the odd numbers from 13 to 835 = 424 Answer
Method (2) to find the average of the odd numbers from 13 to 835
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 835 are
13, 15, 17, . . . . 835
The odd numbers from 13 to 835 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 835
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 835
835 = 13 + (n – 1) × 2
⇒ 835 = 13 + 2 n – 2
⇒ 835 = 13 – 2 + 2 n
⇒ 835 = 11 + 2 n
After transposing 11 to LHS
⇒ 835 – 11 = 2 n
⇒ 824 = 2 n
After rearranging the above expression
⇒ 2 n = 824
After transposing 2 to RHS
⇒ n = 824/2
⇒ n = 412
Thus, the number of terms of odd numbers from 13 to 835 = 412
This means 835 is the 412th term.
Finding the sum of the given odd numbers from 13 to 835
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 835
= 412/2 (13 + 835)
= 412/2 × 848
= 412 × 848/2
= 349376/2 = 174688
Thus, the sum of all terms of the given odd numbers from 13 to 835 = 174688
And, the total number of terms = 412
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 835
= 174688/412 = 424
Thus, the average of the given odd numbers from 13 to 835 = 424 Answer
Similar Questions
(1) Find the average of odd numbers from 5 to 189
(2) Find the average of the first 3018 even numbers.
(3) Find the average of the first 4851 even numbers.
(4) Find the average of even numbers from 10 to 1330
(5) Find the average of the first 627 odd numbers.
(6) Find the average of the first 1214 odd numbers.
(7) Find the average of even numbers from 4 to 484
(8) Find the average of even numbers from 4 to 278
(9) What is the average of the first 1979 even numbers?
(10) Find the average of even numbers from 10 to 422