Question:
Find the average of odd numbers from 15 to 969
Correct Answer
492
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 969
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 969 are
15, 17, 19, . . . . 969
After observing the above list of the odd numbers from 15 to 969 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 969 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 969
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 969
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 969
= 15 + 969/2
= 984/2 = 492
Thus, the average of the odd numbers from 15 to 969 = 492 Answer
Method (2) to find the average of the odd numbers from 15 to 969
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 969 are
15, 17, 19, . . . . 969
The odd numbers from 15 to 969 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 969
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 969
969 = 15 + (n – 1) × 2
⇒ 969 = 15 + 2 n – 2
⇒ 969 = 15 – 2 + 2 n
⇒ 969 = 13 + 2 n
After transposing 13 to LHS
⇒ 969 – 13 = 2 n
⇒ 956 = 2 n
After rearranging the above expression
⇒ 2 n = 956
After transposing 2 to RHS
⇒ n = 956/2
⇒ n = 478
Thus, the number of terms of odd numbers from 15 to 969 = 478
This means 969 is the 478th term.
Finding the sum of the given odd numbers from 15 to 969
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 969
= 478/2 (15 + 969)
= 478/2 × 984
= 478 × 984/2
= 470352/2 = 235176
Thus, the sum of all terms of the given odd numbers from 15 to 969 = 235176
And, the total number of terms = 478
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 969
= 235176/478 = 492
Thus, the average of the given odd numbers from 15 to 969 = 492 Answer
Similar Questions
(1) Find the average of the first 1357 odd numbers.
(2) Find the average of odd numbers from 11 to 1239
(3) Find the average of odd numbers from 5 to 1425
(4) Find the average of even numbers from 4 to 810
(5) What will be the average of the first 4959 odd numbers?
(6) Find the average of the first 1664 odd numbers.
(7) Find the average of the first 1484 odd numbers.
(8) Find the average of odd numbers from 9 to 1335
(9) What is the average of the first 840 even numbers?
(10) Find the average of even numbers from 6 to 818