Question:
Find the average of even numbers from 12 to 678
Correct Answer
345
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 678
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 678 are
12, 14, 16, . . . . 678
After observing the above list of the even numbers from 12 to 678 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 678 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 678
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 678
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 678
= 12 + 678/2
= 690/2 = 345
Thus, the average of the even numbers from 12 to 678 = 345 Answer
Method (2) to find the average of the even numbers from 12 to 678
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 678 are
12, 14, 16, . . . . 678
The even numbers from 12 to 678 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 678
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 678
678 = 12 + (n – 1) × 2
⇒ 678 = 12 + 2 n – 2
⇒ 678 = 12 – 2 + 2 n
⇒ 678 = 10 + 2 n
After transposing 10 to LHS
⇒ 678 – 10 = 2 n
⇒ 668 = 2 n
After rearranging the above expression
⇒ 2 n = 668
After transposing 2 to RHS
⇒ n = 668/2
⇒ n = 334
Thus, the number of terms of even numbers from 12 to 678 = 334
This means 678 is the 334th term.
Finding the sum of the given even numbers from 12 to 678
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 678
= 334/2 (12 + 678)
= 334/2 × 690
= 334 × 690/2
= 230460/2 = 115230
Thus, the sum of all terms of the given even numbers from 12 to 678 = 115230
And, the total number of terms = 334
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 678
= 115230/334 = 345
Thus, the average of the given even numbers from 12 to 678 = 345 Answer
Similar Questions
(1) Find the average of the first 305 odd numbers.
(2) What will be the average of the first 4225 odd numbers?
(3) What will be the average of the first 4294 odd numbers?
(4) Find the average of odd numbers from 3 to 481
(5) Find the average of the first 2264 odd numbers.
(6) Find the average of the first 1581 odd numbers.
(7) Find the average of the first 2368 odd numbers.
(8) Find the average of the first 1847 odd numbers.
(9) Find the average of the first 3759 odd numbers.
(10) Find the average of even numbers from 8 to 1324