Question : Find the average of odd numbers from 15 to 1025
Correct Answer 520
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1025
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1025 are
15, 17, 19, . . . . 1025
After observing the above list of the odd numbers from 15 to 1025 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1025 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1025
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1025
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 1025
= 15 + 1025/2
= 1040/2 = 520
Thus, the average of the odd numbers from 15 to 1025 = 520 Answer
Method (2) to find the average of the odd numbers from 15 to 1025
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1025 are
15, 17, 19, . . . . 1025
The odd numbers from 15 to 1025 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1025
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 1025
1025 = 15 + (n – 1) × 2
⇒ 1025 = 15 + 2 n – 2
⇒ 1025 = 15 – 2 + 2 n
⇒ 1025 = 13 + 2 n
After transposing 13 to LHS
⇒ 1025 – 13 = 2 n
⇒ 1012 = 2 n
After rearranging the above expression
⇒ 2 n = 1012
After transposing 2 to RHS
⇒ n = 1012/2
⇒ n = 506
Thus, the number of terms of odd numbers from 15 to 1025 = 506
This means 1025 is the 506th term.
Finding the sum of the given odd numbers from 15 to 1025
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 1025
= 506/2 (15 + 1025)
= 506/2 × 1040
= 506 × 1040/2
= 526240/2 = 263120
Thus, the sum of all terms of the given odd numbers from 15 to 1025 = 263120
And, the total number of terms = 506
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 1025
= 263120/506 = 520
Thus, the average of the given odd numbers from 15 to 1025 = 520 Answer
Similar Questions
(1) What is the average of the first 182 even numbers?
(2) Find the average of even numbers from 12 to 352
(3) What will be the average of the first 4558 odd numbers?
(4) Find the average of even numbers from 8 to 1114
(5) Find the average of the first 645 odd numbers.
(6) What is the average of the first 579 even numbers?
(7) Find the average of even numbers from 8 to 828
(8) Find the average of even numbers from 10 to 818
(9) Find the average of the first 2733 odd numbers.
(10) Find the average of the first 1727 odd numbers.