Question:
Find the average of odd numbers from 15 to 949
Correct Answer
482
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 949
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 949 are
15, 17, 19, . . . . 949
After observing the above list of the odd numbers from 15 to 949 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 949 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 949
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 949
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 15 to 949
= 15 + 949/2
= 964/2 = 482
Thus, the average of the odd numbers from 15 to 949 = 482 Answer
Method (2) to find the average of the odd numbers from 15 to 949
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 949 are
15, 17, 19, . . . . 949
The odd numbers from 15 to 949 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 949
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 15 to 949
949 = 15 + (n – 1) × 2
⇒ 949 = 15 + 2 n – 2
⇒ 949 = 15 – 2 + 2 n
⇒ 949 = 13 + 2 n
After transposing 13 to LHS
⇒ 949 – 13 = 2 n
⇒ 936 = 2 n
After rearranging the above expression
⇒ 2 n = 936
After transposing 2 to RHS
⇒ n = 936/2
⇒ n = 468
Thus, the number of terms of odd numbers from 15 to 949 = 468
This means 949 is the 468th term.
Finding the sum of the given odd numbers from 15 to 949
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 15 to 949
= 468/2 (15 + 949)
= 468/2 × 964
= 468 × 964/2
= 451152/2 = 225576
Thus, the sum of all terms of the given odd numbers from 15 to 949 = 225576
And, the total number of terms = 468
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 15 to 949
= 225576/468 = 482
Thus, the average of the given odd numbers from 15 to 949 = 482 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 1275
(2) Find the average of the first 2448 even numbers.
(3) Find the average of the first 1474 odd numbers.
(4) Find the average of odd numbers from 11 to 813
(5) Find the average of the first 3212 odd numbers.
(6) Find the average of odd numbers from 13 to 1265
(7) Find the average of odd numbers from 13 to 1145
(8) Find the average of odd numbers from 5 to 1417
(9) Find the average of odd numbers from 11 to 581
(10) Find the average of the first 904 odd numbers.