Question : Find the average of odd numbers from 7 to 657
Correct Answer 332
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 657
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 657 are
7, 9, 11, . . . . 657
After observing the above list of the odd numbers from 7 to 657 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 657 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 657
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 657
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 657
= 7 + 657/2
= 664/2 = 332
Thus, the average of the odd numbers from 7 to 657 = 332 Answer
Method (2) to find the average of the odd numbers from 7 to 657
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 657 are
7, 9, 11, . . . . 657
The odd numbers from 7 to 657 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 657
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 657
657 = 7 + (n – 1) × 2
⇒ 657 = 7 + 2 n – 2
⇒ 657 = 7 – 2 + 2 n
⇒ 657 = 5 + 2 n
After transposing 5 to LHS
⇒ 657 – 5 = 2 n
⇒ 652 = 2 n
After rearranging the above expression
⇒ 2 n = 652
After transposing 2 to RHS
⇒ n = 652/2
⇒ n = 326
Thus, the number of terms of odd numbers from 7 to 657 = 326
This means 657 is the 326th term.
Finding the sum of the given odd numbers from 7 to 657
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 657
= 326/2 (7 + 657)
= 326/2 × 664
= 326 × 664/2
= 216464/2 = 108232
Thus, the sum of all terms of the given odd numbers from 7 to 657 = 108232
And, the total number of terms = 326
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 657
= 108232/326 = 332
Thus, the average of the given odd numbers from 7 to 657 = 332 Answer
Similar Questions
(1) Find the average of the first 2347 even numbers.
(2) Find the average of the first 506 odd numbers.
(3) What is the average of the first 1306 even numbers?
(4) Find the average of the first 1039 odd numbers.
(5) What will be the average of the first 4164 odd numbers?
(6) Find the average of even numbers from 10 to 1464
(7) Find the average of even numbers from 8 to 1246
(8) Find the average of even numbers from 12 to 284
(9) What is the average of the first 470 even numbers?
(10) Find the average of first 30 multiples of 11.