Question:
Find the average of odd numbers from 13 to 393
Correct Answer
203
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 393
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 393 are
13, 15, 17, . . . . 393
After observing the above list of the odd numbers from 13 to 393 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 393 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 393
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 393
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 393
= 13 + 393/2
= 406/2 = 203
Thus, the average of the odd numbers from 13 to 393 = 203 Answer
Method (2) to find the average of the odd numbers from 13 to 393
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 393 are
13, 15, 17, . . . . 393
The odd numbers from 13 to 393 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 393
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 393
393 = 13 + (n – 1) × 2
⇒ 393 = 13 + 2 n – 2
⇒ 393 = 13 – 2 + 2 n
⇒ 393 = 11 + 2 n
After transposing 11 to LHS
⇒ 393 – 11 = 2 n
⇒ 382 = 2 n
After rearranging the above expression
⇒ 2 n = 382
After transposing 2 to RHS
⇒ n = 382/2
⇒ n = 191
Thus, the number of terms of odd numbers from 13 to 393 = 191
This means 393 is the 191th term.
Finding the sum of the given odd numbers from 13 to 393
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 393
= 191/2 (13 + 393)
= 191/2 × 406
= 191 × 406/2
= 77546/2 = 38773
Thus, the sum of all terms of the given odd numbers from 13 to 393 = 38773
And, the total number of terms = 191
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 393
= 38773/191 = 203
Thus, the average of the given odd numbers from 13 to 393 = 203 Answer
Similar Questions
(1) Find the average of the first 3695 even numbers.
(2) What is the average of the first 28 even numbers?
(3) Find the average of the first 3162 odd numbers.
(4) Find the average of even numbers from 4 to 1212
(5) Find the average of the first 2391 odd numbers.
(6) Find the average of even numbers from 8 to 514
(7) Find the average of the first 3549 even numbers.
(8) What is the average of the first 1654 even numbers?
(9) Find the average of odd numbers from 5 to 1183
(10) Find the average of the first 1177 odd numbers.