Question:
Find the average of odd numbers from 7 to 1275
Correct Answer
641
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 1275
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 1275 are
7, 9, 11, . . . . 1275
After observing the above list of the odd numbers from 7 to 1275 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 1275 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 1275
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 1275
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 1275
= 7 + 1275/2
= 1282/2 = 641
Thus, the average of the odd numbers from 7 to 1275 = 641 Answer
Method (2) to find the average of the odd numbers from 7 to 1275
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 1275 are
7, 9, 11, . . . . 1275
The odd numbers from 7 to 1275 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 1275
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 1275
1275 = 7 + (n – 1) × 2
⇒ 1275 = 7 + 2 n – 2
⇒ 1275 = 7 – 2 + 2 n
⇒ 1275 = 5 + 2 n
After transposing 5 to LHS
⇒ 1275 – 5 = 2 n
⇒ 1270 = 2 n
After rearranging the above expression
⇒ 2 n = 1270
After transposing 2 to RHS
⇒ n = 1270/2
⇒ n = 635
Thus, the number of terms of odd numbers from 7 to 1275 = 635
This means 1275 is the 635th term.
Finding the sum of the given odd numbers from 7 to 1275
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 1275
= 635/2 (7 + 1275)
= 635/2 × 1282
= 635 × 1282/2
= 814070/2 = 407035
Thus, the sum of all terms of the given odd numbers from 7 to 1275 = 407035
And, the total number of terms = 635
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 1275
= 407035/635 = 641
Thus, the average of the given odd numbers from 7 to 1275 = 641 Answer
Similar Questions
(1) Find the average of the first 1139 odd numbers.
(2) Find the average of the first 3172 odd numbers.
(3) Find the average of the first 1799 odd numbers.
(4) Find the average of the first 2439 even numbers.
(5) Find the average of the first 3617 even numbers.
(6) What is the average of the first 461 even numbers?
(7) What is the average of the first 218 even numbers?
(8) What will be the average of the first 4655 odd numbers?
(9) Find the average of even numbers from 10 to 54
(10) Find the average of even numbers from 10 to 1088