Question:
Find the average of even numbers from 4 to 430
Correct Answer
217
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 430
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 430 are
4, 6, 8, . . . . 430
After observing the above list of the even numbers from 4 to 430 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 430 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 430
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 430
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 430
= 4 + 430/2
= 434/2 = 217
Thus, the average of the even numbers from 4 to 430 = 217 Answer
Method (2) to find the average of the even numbers from 4 to 430
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 430 are
4, 6, 8, . . . . 430
The even numbers from 4 to 430 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 430
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 430
430 = 4 + (n – 1) × 2
⇒ 430 = 4 + 2 n – 2
⇒ 430 = 4 – 2 + 2 n
⇒ 430 = 2 + 2 n
After transposing 2 to LHS
⇒ 430 – 2 = 2 n
⇒ 428 = 2 n
After rearranging the above expression
⇒ 2 n = 428
After transposing 2 to RHS
⇒ n = 428/2
⇒ n = 214
Thus, the number of terms of even numbers from 4 to 430 = 214
This means 430 is the 214th term.
Finding the sum of the given even numbers from 4 to 430
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 430
= 214/2 (4 + 430)
= 214/2 × 434
= 214 × 434/2
= 92876/2 = 46438
Thus, the sum of all terms of the given even numbers from 4 to 430 = 46438
And, the total number of terms = 214
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 430
= 46438/214 = 217
Thus, the average of the given even numbers from 4 to 430 = 217 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1018
(2) Find the average of even numbers from 12 to 1454
(3) What is the average of the first 213 even numbers?
(4) Find the average of odd numbers from 9 to 691
(5) Find the average of the first 1426 odd numbers.
(6) Find the average of odd numbers from 15 to 569
(7) Find the average of the first 2425 odd numbers.
(8) Find the average of the first 3380 even numbers.
(9) Find the average of odd numbers from 15 to 1151
(10) Find the average of even numbers from 6 to 1770