Question:
Find the average of even numbers from 12 to 1414
Correct Answer
713
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1414
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1414 are
12, 14, 16, . . . . 1414
After observing the above list of the even numbers from 12 to 1414 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1414 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1414
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1414
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 1414
= 12 + 1414/2
= 1426/2 = 713
Thus, the average of the even numbers from 12 to 1414 = 713 Answer
Method (2) to find the average of the even numbers from 12 to 1414
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1414 are
12, 14, 16, . . . . 1414
The even numbers from 12 to 1414 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1414
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 1414
1414 = 12 + (n – 1) × 2
⇒ 1414 = 12 + 2 n – 2
⇒ 1414 = 12 – 2 + 2 n
⇒ 1414 = 10 + 2 n
After transposing 10 to LHS
⇒ 1414 – 10 = 2 n
⇒ 1404 = 2 n
After rearranging the above expression
⇒ 2 n = 1404
After transposing 2 to RHS
⇒ n = 1404/2
⇒ n = 702
Thus, the number of terms of even numbers from 12 to 1414 = 702
This means 1414 is the 702th term.
Finding the sum of the given even numbers from 12 to 1414
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 1414
= 702/2 (12 + 1414)
= 702/2 × 1426
= 702 × 1426/2
= 1001052/2 = 500526
Thus, the sum of all terms of the given even numbers from 12 to 1414 = 500526
And, the total number of terms = 702
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 1414
= 500526/702 = 713
Thus, the average of the given even numbers from 12 to 1414 = 713 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 1234
(2) Find the average of the first 3110 even numbers.
(3) Find the average of the first 1154 odd numbers.
(4) Find the average of odd numbers from 15 to 1041
(5) Find the average of odd numbers from 15 to 383
(6) What will be the average of the first 4008 odd numbers?
(7) Find the average of even numbers from 10 to 1468
(8) Find the average of odd numbers from 13 to 727
(9) What is the average of the first 1072 even numbers?
(10) Find the average of even numbers from 4 to 1762