Question:
Find the average of odd numbers from 5 to 853
Correct Answer
429
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 853
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 853 are
5, 7, 9, . . . . 853
After observing the above list of the odd numbers from 5 to 853 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 853 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 853
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 853
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 5 to 853
= 5 + 853/2
= 858/2 = 429
Thus, the average of the odd numbers from 5 to 853 = 429 Answer
Method (2) to find the average of the odd numbers from 5 to 853
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 853 are
5, 7, 9, . . . . 853
The odd numbers from 5 to 853 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 853
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 5 to 853
853 = 5 + (n – 1) × 2
⇒ 853 = 5 + 2 n – 2
⇒ 853 = 5 – 2 + 2 n
⇒ 853 = 3 + 2 n
After transposing 3 to LHS
⇒ 853 – 3 = 2 n
⇒ 850 = 2 n
After rearranging the above expression
⇒ 2 n = 850
After transposing 2 to RHS
⇒ n = 850/2
⇒ n = 425
Thus, the number of terms of odd numbers from 5 to 853 = 425
This means 853 is the 425th term.
Finding the sum of the given odd numbers from 5 to 853
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 5 to 853
= 425/2 (5 + 853)
= 425/2 × 858
= 425 × 858/2
= 364650/2 = 182325
Thus, the sum of all terms of the given odd numbers from 5 to 853 = 182325
And, the total number of terms = 425
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 5 to 853
= 182325/425 = 429
Thus, the average of the given odd numbers from 5 to 853 = 429 Answer
Similar Questions
(1) Find the average of the first 2906 even numbers.
(2) Find the average of odd numbers from 9 to 43
(3) Find the average of the first 3843 even numbers.
(4) Find the average of even numbers from 8 to 1424
(5) Find the average of the first 3253 odd numbers.
(6) Find the average of odd numbers from 15 to 471
(7) Find the average of the first 436 odd numbers.
(8) What is the average of the first 1085 even numbers?
(9) Find the average of odd numbers from 11 to 695
(10) Find the average of even numbers from 6 to 1168