Question:
Find the average of even numbers from 12 to 1406
Correct Answer
709
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1406
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1406 are
12, 14, 16, . . . . 1406
After observing the above list of the even numbers from 12 to 1406 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1406 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1406
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1406
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 1406
= 12 + 1406/2
= 1418/2 = 709
Thus, the average of the even numbers from 12 to 1406 = 709 Answer
Method (2) to find the average of the even numbers from 12 to 1406
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1406 are
12, 14, 16, . . . . 1406
The even numbers from 12 to 1406 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1406
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 1406
1406 = 12 + (n – 1) × 2
⇒ 1406 = 12 + 2 n – 2
⇒ 1406 = 12 – 2 + 2 n
⇒ 1406 = 10 + 2 n
After transposing 10 to LHS
⇒ 1406 – 10 = 2 n
⇒ 1396 = 2 n
After rearranging the above expression
⇒ 2 n = 1396
After transposing 2 to RHS
⇒ n = 1396/2
⇒ n = 698
Thus, the number of terms of even numbers from 12 to 1406 = 698
This means 1406 is the 698th term.
Finding the sum of the given even numbers from 12 to 1406
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 1406
= 698/2 (12 + 1406)
= 698/2 × 1418
= 698 × 1418/2
= 989764/2 = 494882
Thus, the sum of all terms of the given even numbers from 12 to 1406 = 494882
And, the total number of terms = 698
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 1406
= 494882/698 = 709
Thus, the average of the given even numbers from 12 to 1406 = 709 Answer
Similar Questions
(1) Find the average of the first 648 odd numbers.
(2) Find the average of the first 2014 even numbers.
(3) Find the average of the first 640 odd numbers.
(4) Find the average of the first 3618 even numbers.
(5) Find the average of odd numbers from 5 to 1003
(6) Find the average of odd numbers from 5 to 1033
(7) Find the average of the first 2594 odd numbers.
(8) What is the average of the first 885 even numbers?
(9) Find the average of even numbers from 6 to 820
(10) Find the average of the first 1140 odd numbers.