Question : Find the average of even numbers from 6 to 414
Correct Answer 210
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 414
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 414 are
6, 8, 10, . . . . 414
After observing the above list of the even numbers from 6 to 414 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 414 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 414
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 414
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 6 to 414
= 6 + 414/2
= 420/2 = 210
Thus, the average of the even numbers from 6 to 414 = 210 Answer
Method (2) to find the average of the even numbers from 6 to 414
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 414 are
6, 8, 10, . . . . 414
The even numbers from 6 to 414 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 414
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 6 to 414
414 = 6 + (n – 1) × 2
⇒ 414 = 6 + 2 n – 2
⇒ 414 = 6 – 2 + 2 n
⇒ 414 = 4 + 2 n
After transposing 4 to LHS
⇒ 414 – 4 = 2 n
⇒ 410 = 2 n
After rearranging the above expression
⇒ 2 n = 410
After transposing 2 to RHS
⇒ n = 410/2
⇒ n = 205
Thus, the number of terms of even numbers from 6 to 414 = 205
This means 414 is the 205th term.
Finding the sum of the given even numbers from 6 to 414
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 6 to 414
= 205/2 (6 + 414)
= 205/2 × 420
= 205 × 420/2
= 86100/2 = 43050
Thus, the sum of all terms of the given even numbers from 6 to 414 = 43050
And, the total number of terms = 205
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 6 to 414
= 43050/205 = 210
Thus, the average of the given even numbers from 6 to 414 = 210 Answer
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