Question:
Find the average of odd numbers from 13 to 659
Correct Answer
336
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 659
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 659 are
13, 15, 17, . . . . 659
After observing the above list of the odd numbers from 13 to 659 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 659 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 659
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 659
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 659
= 13 + 659/2
= 672/2 = 336
Thus, the average of the odd numbers from 13 to 659 = 336 Answer
Method (2) to find the average of the odd numbers from 13 to 659
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 659 are
13, 15, 17, . . . . 659
The odd numbers from 13 to 659 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 659
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 659
659 = 13 + (n – 1) × 2
⇒ 659 = 13 + 2 n – 2
⇒ 659 = 13 – 2 + 2 n
⇒ 659 = 11 + 2 n
After transposing 11 to LHS
⇒ 659 – 11 = 2 n
⇒ 648 = 2 n
After rearranging the above expression
⇒ 2 n = 648
After transposing 2 to RHS
⇒ n = 648/2
⇒ n = 324
Thus, the number of terms of odd numbers from 13 to 659 = 324
This means 659 is the 324th term.
Finding the sum of the given odd numbers from 13 to 659
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 659
= 324/2 (13 + 659)
= 324/2 × 672
= 324 × 672/2
= 217728/2 = 108864
Thus, the sum of all terms of the given odd numbers from 13 to 659 = 108864
And, the total number of terms = 324
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 659
= 108864/324 = 336
Thus, the average of the given odd numbers from 13 to 659 = 336 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 491
(2) Find the average of odd numbers from 3 to 455
(3) Find the average of odd numbers from 5 to 897
(4) Find the average of even numbers from 12 to 1922
(5) Find the average of odd numbers from 13 to 1407
(6) Find the average of the first 1391 odd numbers.
(7) Find the average of the first 3177 even numbers.
(8) Find the average of odd numbers from 11 to 577
(9) Find the average of odd numbers from 9 to 1077
(10) Find the average of odd numbers from 15 to 823