Question:
Find the average of odd numbers from 9 to 895
Correct Answer
452
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 895
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 895 are
9, 11, 13, . . . . 895
After observing the above list of the odd numbers from 9 to 895 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 895 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 895
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 895
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 9 to 895
= 9 + 895/2
= 904/2 = 452
Thus, the average of the odd numbers from 9 to 895 = 452 Answer
Method (2) to find the average of the odd numbers from 9 to 895
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 895 are
9, 11, 13, . . . . 895
The odd numbers from 9 to 895 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 895
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 9 to 895
895 = 9 + (n – 1) × 2
⇒ 895 = 9 + 2 n – 2
⇒ 895 = 9 – 2 + 2 n
⇒ 895 = 7 + 2 n
After transposing 7 to LHS
⇒ 895 – 7 = 2 n
⇒ 888 = 2 n
After rearranging the above expression
⇒ 2 n = 888
After transposing 2 to RHS
⇒ n = 888/2
⇒ n = 444
Thus, the number of terms of odd numbers from 9 to 895 = 444
This means 895 is the 444th term.
Finding the sum of the given odd numbers from 9 to 895
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 9 to 895
= 444/2 (9 + 895)
= 444/2 × 904
= 444 × 904/2
= 401376/2 = 200688
Thus, the sum of all terms of the given odd numbers from 9 to 895 = 200688
And, the total number of terms = 444
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 9 to 895
= 200688/444 = 452
Thus, the average of the given odd numbers from 9 to 895 = 452 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 1031
(2) Find the average of even numbers from 10 to 540
(3) Find the average of odd numbers from 9 to 933
(4) Find the average of the first 3375 even numbers.
(5) Find the average of even numbers from 12 to 1404
(6) Find the average of odd numbers from 15 to 375
(7) Find the average of even numbers from 6 to 24
(8) Find the average of the first 2667 odd numbers.
(9) Find the average of odd numbers from 7 to 265
(10) What is the average of the first 62 odd numbers?