Question:
Find the average of odd numbers from 15 to 893
Correct Answer
454
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 893
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 893 are
15, 17, 19, . . . . 893
After observing the above list of the odd numbers from 15 to 893 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 893 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 893
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 893
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 893
= 15 + 893/2
= 908/2 = 454
Thus, the average of the odd numbers from 15 to 893 = 454 Answer
Method (2) to find the average of the odd numbers from 15 to 893
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 893 are
15, 17, 19, . . . . 893
The odd numbers from 15 to 893 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 893
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 893
893 = 15 + (n – 1) × 2
⇒ 893 = 15 + 2 n – 2
⇒ 893 = 15 – 2 + 2 n
⇒ 893 = 13 + 2 n
After transposing 13 to LHS
⇒ 893 – 13 = 2 n
⇒ 880 = 2 n
After rearranging the above expression
⇒ 2 n = 880
After transposing 2 to RHS
⇒ n = 880/2
⇒ n = 440
Thus, the number of terms of odd numbers from 15 to 893 = 440
This means 893 is the 440th term.
Finding the sum of the given odd numbers from 15 to 893
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 893
= 440/2 (15 + 893)
= 440/2 × 908
= 440 × 908/2
= 399520/2 = 199760
Thus, the sum of all terms of the given odd numbers from 15 to 893 = 199760
And, the total number of terms = 440
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 893
= 199760/440 = 454
Thus, the average of the given odd numbers from 15 to 893 = 454 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 372
(2) What is the average of the first 707 even numbers?
(3) Find the average of odd numbers from 5 to 1443
(4) Find the average of even numbers from 6 to 1100
(5) What is the average of the first 1076 even numbers?
(6) Find the average of odd numbers from 3 to 1305
(7) Find the average of the first 339 odd numbers.
(8) Find the average of even numbers from 10 to 254
(9) Find the average of odd numbers from 7 to 103
(10) Find the average of even numbers from 4 to 178