Question : Find the average of even numbers from 12 to 698
Correct Answer 355
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 698
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 698 are
12, 14, 16, . . . . 698
After observing the above list of the even numbers from 12 to 698 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 698 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 698
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 698
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 698
= 12 + 698/2
= 710/2 = 355
Thus, the average of the even numbers from 12 to 698 = 355 Answer
Method (2) to find the average of the even numbers from 12 to 698
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 698 are
12, 14, 16, . . . . 698
The even numbers from 12 to 698 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 698
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 698
698 = 12 + (n – 1) × 2
⇒ 698 = 12 + 2 n – 2
⇒ 698 = 12 – 2 + 2 n
⇒ 698 = 10 + 2 n
After transposing 10 to LHS
⇒ 698 – 10 = 2 n
⇒ 688 = 2 n
After rearranging the above expression
⇒ 2 n = 688
After transposing 2 to RHS
⇒ n = 688/2
⇒ n = 344
Thus, the number of terms of even numbers from 12 to 698 = 344
This means 698 is the 344th term.
Finding the sum of the given even numbers from 12 to 698
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 698
= 344/2 (12 + 698)
= 344/2 × 710
= 344 × 710/2
= 244240/2 = 122120
Thus, the sum of all terms of the given even numbers from 12 to 698 = 122120
And, the total number of terms = 344
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 698
= 122120/344 = 355
Thus, the average of the given even numbers from 12 to 698 = 355 Answer
Similar Questions
(1) What is the average of the first 1573 even numbers?
(2) Find the average of odd numbers from 11 to 901
(3) Find the average of the first 3256 even numbers.
(4) What will be the average of the first 4337 odd numbers?
(5) Find the average of odd numbers from 5 to 1243
(6) Find the average of the first 2812 even numbers.
(7) Find the average of even numbers from 6 to 210
(8) Find the average of even numbers from 12 to 1480
(9) Find the average of the first 3975 even numbers.
(10) Find the average of the first 4347 even numbers.