Question : Find the average of even numbers from 12 to 1658
Correct Answer 835
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1658
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1658 are
12, 14, 16, . . . . 1658
After observing the above list of the even numbers from 12 to 1658 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1658 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1658
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1658
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 1658
= 12 + 1658/2
= 1670/2 = 835
Thus, the average of the even numbers from 12 to 1658 = 835 Answer
Method (2) to find the average of the even numbers from 12 to 1658
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1658 are
12, 14, 16, . . . . 1658
The even numbers from 12 to 1658 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1658
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 1658
1658 = 12 + (n – 1) × 2
⇒ 1658 = 12 + 2 n – 2
⇒ 1658 = 12 – 2 + 2 n
⇒ 1658 = 10 + 2 n
After transposing 10 to LHS
⇒ 1658 – 10 = 2 n
⇒ 1648 = 2 n
After rearranging the above expression
⇒ 2 n = 1648
After transposing 2 to RHS
⇒ n = 1648/2
⇒ n = 824
Thus, the number of terms of even numbers from 12 to 1658 = 824
This means 1658 is the 824th term.
Finding the sum of the given even numbers from 12 to 1658
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 1658
= 824/2 (12 + 1658)
= 824/2 × 1670
= 824 × 1670/2
= 1376080/2 = 688040
Thus, the sum of all terms of the given even numbers from 12 to 1658 = 688040
And, the total number of terms = 824
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 1658
= 688040/824 = 835
Thus, the average of the given even numbers from 12 to 1658 = 835 Answer
Similar Questions
(1) What is the average of the first 990 even numbers?
(2) Find the average of the first 3828 even numbers.
(3) Find the average of the first 5000 even numbers.
(4) Find the average of even numbers from 8 to 442
(5) Find the average of odd numbers from 15 to 1103
(6) Find the average of the first 2535 even numbers.
(7) Find the average of even numbers from 10 to 1050
(8) Find the average of the first 3173 odd numbers.
(9) Find the average of even numbers from 10 to 254
(10) Find the average of even numbers from 12 to 1702