Question:
Find the average of odd numbers from 13 to 285
Correct Answer
149
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 285
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 285 are
13, 15, 17, . . . . 285
After observing the above list of the odd numbers from 13 to 285 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 285 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 285
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 285
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 285
= 13 + 285/2
= 298/2 = 149
Thus, the average of the odd numbers from 13 to 285 = 149 Answer
Method (2) to find the average of the odd numbers from 13 to 285
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 285 are
13, 15, 17, . . . . 285
The odd numbers from 13 to 285 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 285
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 285
285 = 13 + (n – 1) × 2
⇒ 285 = 13 + 2 n – 2
⇒ 285 = 13 – 2 + 2 n
⇒ 285 = 11 + 2 n
After transposing 11 to LHS
⇒ 285 – 11 = 2 n
⇒ 274 = 2 n
After rearranging the above expression
⇒ 2 n = 274
After transposing 2 to RHS
⇒ n = 274/2
⇒ n = 137
Thus, the number of terms of odd numbers from 13 to 285 = 137
This means 285 is the 137th term.
Finding the sum of the given odd numbers from 13 to 285
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 285
= 137/2 (13 + 285)
= 137/2 × 298
= 137 × 298/2
= 40826/2 = 20413
Thus, the sum of all terms of the given odd numbers from 13 to 285 = 20413
And, the total number of terms = 137
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 285
= 20413/137 = 149
Thus, the average of the given odd numbers from 13 to 285 = 149 Answer
Similar Questions
(1) What will be the average of the first 4192 odd numbers?
(2) Find the average of the first 1181 odd numbers.
(3) Find the average of odd numbers from 3 to 657
(4) Find the average of even numbers from 6 to 1818
(5) What is the average of the first 1338 even numbers?
(6) Find the average of even numbers from 12 to 1874
(7) Find the average of the first 3087 odd numbers.
(8) What is the average of the first 792 even numbers?
(9) What will be the average of the first 4643 odd numbers?
(10) Find the average of odd numbers from 7 to 609