Question:
Find the average of even numbers from 12 to 410
Correct Answer
211
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 410
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 410 are
12, 14, 16, . . . . 410
After observing the above list of the even numbers from 12 to 410 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 410 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 410
The First Term (a) = 12
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 12 to 410
= 12 + 410/2
= 422/2 = 211
Thus, the average of the even numbers from 12 to 410 = 211 Answer
Method (2) to find the average of the even numbers from 12 to 410
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 410 are
12, 14, 16, . . . . 410
The even numbers from 12 to 410 form an Arithmetic Series in which
The First Term (a) = 12
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 12 to 410
410 = 12 + (n – 1) × 2
⇒ 410 = 12 + 2 n – 2
⇒ 410 = 12 – 2 + 2 n
⇒ 410 = 10 + 2 n
After transposing 10 to LHS
⇒ 410 – 10 = 2 n
⇒ 400 = 2 n
After rearranging the above expression
⇒ 2 n = 400
After transposing 2 to RHS
⇒ n = 400/2
⇒ n = 200
Thus, the number of terms of even numbers from 12 to 410 = 200
This means 410 is the 200th term.
Finding the sum of the given even numbers from 12 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 12 to 410
= 200/2 (12 + 410)
= 200/2 × 422
= 200 × 422/2
= 84400/2 = 42200
Thus, the sum of all terms of the given even numbers from 12 to 410 = 42200
And, the total number of terms = 200
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 12 to 410
= 42200/200 = 211
Thus, the average of the given even numbers from 12 to 410 = 211 Answer
Similar Questions
(1) Find the average of the first 3694 even numbers.
(2) Find the average of even numbers from 10 to 1826
(3) Find the average of even numbers from 4 to 1288
(4) Find the average of the first 365 odd numbers.
(5) Find the average of even numbers from 4 to 1196
(6) Find the average of the first 3446 odd numbers.
(7) Find the average of odd numbers from 5 to 893
(8) Find the average of odd numbers from 7 to 639
(9) Find the average of even numbers from 10 to 94
(10) Find the average of even numbers from 12 to 1716