Question:
Find the average of odd numbers from 13 to 595
Correct Answer
304
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 595
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 595 are
13, 15, 17, . . . . 595
After observing the above list of the odd numbers from 13 to 595 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 595 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 595
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 595
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 595
= 13 + 595/2
= 608/2 = 304
Thus, the average of the odd numbers from 13 to 595 = 304 Answer
Method (2) to find the average of the odd numbers from 13 to 595
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 595 are
13, 15, 17, . . . . 595
The odd numbers from 13 to 595 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 595
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 595
595 = 13 + (n – 1) × 2
⇒ 595 = 13 + 2 n – 2
⇒ 595 = 13 – 2 + 2 n
⇒ 595 = 11 + 2 n
After transposing 11 to LHS
⇒ 595 – 11 = 2 n
⇒ 584 = 2 n
After rearranging the above expression
⇒ 2 n = 584
After transposing 2 to RHS
⇒ n = 584/2
⇒ n = 292
Thus, the number of terms of odd numbers from 13 to 595 = 292
This means 595 is the 292th term.
Finding the sum of the given odd numbers from 13 to 595
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 595
= 292/2 (13 + 595)
= 292/2 × 608
= 292 × 608/2
= 177536/2 = 88768
Thus, the sum of all terms of the given odd numbers from 13 to 595 = 88768
And, the total number of terms = 292
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 595
= 88768/292 = 304
Thus, the average of the given odd numbers from 13 to 595 = 304 Answer
Similar Questions
(1) Find the average of the first 2327 even numbers.
(2) Find the average of even numbers from 10 to 992
(3) Find the average of the first 4867 even numbers.
(4) Find the average of even numbers from 6 to 736
(5) Find the average of odd numbers from 11 to 1271
(6) Find the average of even numbers from 8 to 750
(7) Find the average of odd numbers from 13 to 933
(8) Find the average of odd numbers from 13 to 1271
(9) Find the average of even numbers from 8 to 786
(10) Find the average of even numbers from 8 to 1062