Question:
Find the average of even numbers from 4 to 214
Correct Answer
109
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 214
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 214 are
4, 6, 8, . . . . 214
After observing the above list of the even numbers from 4 to 214 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 214 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 214
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 214
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 214
= 4 + 214/2
= 218/2 = 109
Thus, the average of the even numbers from 4 to 214 = 109 Answer
Method (2) to find the average of the even numbers from 4 to 214
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 214 are
4, 6, 8, . . . . 214
The even numbers from 4 to 214 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 214
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 214
214 = 4 + (n – 1) × 2
⇒ 214 = 4 + 2 n – 2
⇒ 214 = 4 – 2 + 2 n
⇒ 214 = 2 + 2 n
After transposing 2 to LHS
⇒ 214 – 2 = 2 n
⇒ 212 = 2 n
After rearranging the above expression
⇒ 2 n = 212
After transposing 2 to RHS
⇒ n = 212/2
⇒ n = 106
Thus, the number of terms of even numbers from 4 to 214 = 106
This means 214 is the 106th term.
Finding the sum of the given even numbers from 4 to 214
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 214
= 106/2 (4 + 214)
= 106/2 × 218
= 106 × 218/2
= 23108/2 = 11554
Thus, the sum of all terms of the given even numbers from 4 to 214 = 11554
And, the total number of terms = 106
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 214
= 11554/106 = 109
Thus, the average of the given even numbers from 4 to 214 = 109 Answer
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