Question : Find the average of even numbers from 12 to 1648
Correct Answer 830
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1648
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1648 are
12, 14, 16, . . . . 1648
After observing the above list of the even numbers from 12 to 1648 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1648 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1648
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1648
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 1648
= 12 + 1648/2
= 1660/2 = 830
Thus, the average of the even numbers from 12 to 1648 = 830 Answer
Method (2) to find the average of the even numbers from 12 to 1648
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1648 are
12, 14, 16, . . . . 1648
The even numbers from 12 to 1648 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1648
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 1648
1648 = 12 + (n – 1) × 2
⇒ 1648 = 12 + 2 n – 2
⇒ 1648 = 12 – 2 + 2 n
⇒ 1648 = 10 + 2 n
After transposing 10 to LHS
⇒ 1648 – 10 = 2 n
⇒ 1638 = 2 n
After rearranging the above expression
⇒ 2 n = 1638
After transposing 2 to RHS
⇒ n = 1638/2
⇒ n = 819
Thus, the number of terms of even numbers from 12 to 1648 = 819
This means 1648 is the 819th term.
Finding the sum of the given even numbers from 12 to 1648
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 1648
= 819/2 (12 + 1648)
= 819/2 × 1660
= 819 × 1660/2
= 1359540/2 = 679770
Thus, the sum of all terms of the given even numbers from 12 to 1648 = 679770
And, the total number of terms = 819
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 1648
= 679770/819 = 830
Thus, the average of the given even numbers from 12 to 1648 = 830 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 1948
(2) What will be the average of the first 4543 odd numbers?
(3) Find the average of even numbers from 12 to 902
(4) Find the average of the first 1418 odd numbers.
(5) What will be the average of the first 4247 odd numbers?
(6) Find the average of the first 3549 odd numbers.
(7) Find the average of even numbers from 6 to 1730
(8) Find the average of the first 4215 even numbers.
(9) What is the average of the first 985 even numbers?
(10) Find the average of the first 3164 even numbers.