Question:
Find the average of even numbers from 4 to 226
Correct Answer
115
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 226
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 226 are
4, 6, 8, . . . . 226
After observing the above list of the even numbers from 4 to 226 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 226 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 226
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 226
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 even numbers from 4 to 226
= 4 + 226/2
= 230/2 = 115
Thus, the average of the even numbers from 4 to 226 = 115 Answer
Method (2) to find the average of the even numbers from 4 to 226
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 226 are
4, 6, 8, . . . . 226
The even numbers from 4 to 226 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 226
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 even numbers from 4 to 226
226 = 4 + (n – 1) × 2
⇒ 226 = 4 + 2 n – 2
⇒ 226 = 4 – 2 + 2 n
⇒ 226 = 2 + 2 n
After transposing 2 to LHS
⇒ 226 – 2 = 2 n
⇒ 224 = 2 n
After rearranging the above expression
⇒ 2 n = 224
After transposing 2 to RHS
⇒ n = 224/2
⇒ n = 112
Thus, the number of terms of even numbers from 4 to 226 = 112
This means 226 is the 112th term.
Finding the sum of the given even numbers from 4 to 226
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 even numbers from 4 to 226
= 112/2 (4 + 226)
= 112/2 × 230
= 112 × 230/2
= 25760/2 = 12880
Thus, the sum of all terms of the given even numbers from 4 to 226 = 12880
And, the total number of terms = 112
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given even numbers from 4 to 226
= 12880/112 = 115
Thus, the average of the given even numbers from 4 to 226 = 115 Answer
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