Question : Find the average of odd numbers from 7 to 1009
Correct Answer 508
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 1009
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 1009 are
7, 9, 11, . . . . 1009
After observing the above list of the odd numbers from 7 to 1009 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 1009 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 1009
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 1009
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 7 to 1009
= 7 + 1009/2
= 1016/2 = 508
Thus, the average of the odd numbers from 7 to 1009 = 508 Answer
Method (2) to find the average of the odd numbers from 7 to 1009
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 1009 are
7, 9, 11, . . . . 1009
The odd numbers from 7 to 1009 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 1009
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 7 to 1009
1009 = 7 + (n – 1) × 2
⇒ 1009 = 7 + 2 n – 2
⇒ 1009 = 7 – 2 + 2 n
⇒ 1009 = 5 + 2 n
After transposing 5 to LHS
⇒ 1009 – 5 = 2 n
⇒ 1004 = 2 n
After rearranging the above expression
⇒ 2 n = 1004
After transposing 2 to RHS
⇒ n = 1004/2
⇒ n = 502
Thus, the number of terms of odd numbers from 7 to 1009 = 502
This means 1009 is the 502th term.
Finding the sum of the given odd numbers from 7 to 1009
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 7 to 1009
= 502/2 (7 + 1009)
= 502/2 × 1016
= 502 × 1016/2
= 510032/2 = 255016
Thus, the sum of all terms of the given odd numbers from 7 to 1009 = 255016
And, the total number of terms = 502
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 7 to 1009
= 255016/502 = 508
Thus, the average of the given odd numbers from 7 to 1009 = 508 Answer
Similar Questions
(1) Find the average of the first 2541 even numbers.
(2) Find the average of the first 1049 odd numbers.
(3) Find the average of odd numbers from 7 to 375
(4) Find the average of even numbers from 8 to 1012
(5) Find the average of the first 2452 even numbers.
(6) Find the average of even numbers from 12 to 978
(7) What is the average of the first 1282 even numbers?
(8) Find the average of odd numbers from 15 to 625
(9) Find the average of the first 1673 odd numbers.
(10) Find the average of the first 2598 even numbers.