Question:
Find the average of odd numbers from 13 to 1085
Correct Answer
549
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 1085
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 1085 are
13, 15, 17, . . . . 1085
After observing the above list of the odd numbers from 13 to 1085 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 1085 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 1085
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1085
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 13 to 1085
= 13 + 1085/2
= 1098/2 = 549
Thus, the average of the odd numbers from 13 to 1085 = 549 Answer
Method (2) to find the average of the odd numbers from 13 to 1085
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 1085 are
13, 15, 17, . . . . 1085
The odd numbers from 13 to 1085 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1085
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 13 to 1085
1085 = 13 + (n – 1) × 2
⇒ 1085 = 13 + 2 n – 2
⇒ 1085 = 13 – 2 + 2 n
⇒ 1085 = 11 + 2 n
After transposing 11 to LHS
⇒ 1085 – 11 = 2 n
⇒ 1074 = 2 n
After rearranging the above expression
⇒ 2 n = 1074
After transposing 2 to RHS
⇒ n = 1074/2
⇒ n = 537
Thus, the number of terms of odd numbers from 13 to 1085 = 537
This means 1085 is the 537th term.
Finding the sum of the given odd numbers from 13 to 1085
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 13 to 1085
= 537/2 (13 + 1085)
= 537/2 × 1098
= 537 × 1098/2
= 589626/2 = 294813
Thus, the sum of all terms of the given odd numbers from 13 to 1085 = 294813
And, the total number of terms = 537
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 13 to 1085
= 294813/537 = 549
Thus, the average of the given odd numbers from 13 to 1085 = 549 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 1478
(2) What is the average of the first 748 even numbers?
(3) Find the average of even numbers from 8 to 266
(4) Find the average of the first 1231 odd numbers.
(5) Find the average of even numbers from 12 to 940
(6) Find the average of the first 2408 odd numbers.
(7) Find the average of the first 4744 even numbers.
(8) What is the average of the first 19 odd numbers?
(9) Find the average of even numbers from 8 to 766
(10) Find the average of even numbers from 12 to 1882