Question : Find the average of even numbers from 6 to 410
Correct Answer 208
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 410
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 410 are
6, 8, 10, . . . . 410
After observing the above list of the even numbers from 6 to 410 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 410 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 410
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 410
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 410
= 6 + 410/2
= 416/2 = 208
Thus, the average of the even numbers from 6 to 410 = 208 Answer
Method (2) to find the average of the even numbers from 6 to 410
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 410 are
6, 8, 10, . . . . 410
The even numbers from 6 to 410 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 410
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 410
410 = 6 + (n – 1) × 2
⇒ 410 = 6 + 2 n – 2
⇒ 410 = 6 – 2 + 2 n
⇒ 410 = 4 + 2 n
After transposing 4 to LHS
⇒ 410 – 4 = 2 n
⇒ 406 = 2 n
After rearranging the above expression
⇒ 2 n = 406
After transposing 2 to RHS
⇒ n = 406/2
⇒ n = 203
Thus, the number of terms of even numbers from 6 to 410 = 203
This means 410 is the 203th term.
Finding the sum of the given even numbers from 6 to 410
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 410
= 203/2 (6 + 410)
= 203/2 × 416
= 203 × 416/2
= 84448/2 = 42224
Thus, the sum of all terms of the given even numbers from 6 to 410 = 42224
And, the total number of terms = 203
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 410
= 42224/203 = 208
Thus, the average of the given even numbers from 6 to 410 = 208 Answer
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