Question:
Find the average of odd numbers from 13 to 265
Correct Answer
139
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 265
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 265 are
13, 15, 17, . . . . 265
After observing the above list of the odd numbers from 13 to 265 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 265 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 265
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 265
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 265
= 13 + 265/2
= 278/2 = 139
Thus, the average of the odd numbers from 13 to 265 = 139 Answer
Method (2) to find the average of the odd numbers from 13 to 265
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 265 are
13, 15, 17, . . . . 265
The odd numbers from 13 to 265 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 265
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 265
265 = 13 + (n – 1) × 2
⇒ 265 = 13 + 2 n – 2
⇒ 265 = 13 – 2 + 2 n
⇒ 265 = 11 + 2 n
After transposing 11 to LHS
⇒ 265 – 11 = 2 n
⇒ 254 = 2 n
After rearranging the above expression
⇒ 2 n = 254
After transposing 2 to RHS
⇒ n = 254/2
⇒ n = 127
Thus, the number of terms of odd numbers from 13 to 265 = 127
This means 265 is the 127th term.
Finding the sum of the given odd numbers from 13 to 265
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 265
= 127/2 (13 + 265)
= 127/2 × 278
= 127 × 278/2
= 35306/2 = 17653
Thus, the sum of all terms of the given odd numbers from 13 to 265 = 17653
And, the total number of terms = 127
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 265
= 17653/127 = 139
Thus, the average of the given odd numbers from 13 to 265 = 139 Answer
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