Question : Find the average of even numbers from 12 to 1796
Correct Answer 904
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1796
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1796 are
12, 14, 16, . . . . 1796
After observing the above list of the even numbers from 12 to 1796 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1796 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1796
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1796
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 1796
= 12 + 1796/2
= 1808/2 = 904
Thus, the average of the even numbers from 12 to 1796 = 904 Answer
Method (2) to find the average of the even numbers from 12 to 1796
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1796 are
12, 14, 16, . . . . 1796
The even numbers from 12 to 1796 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1796
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 1796
1796 = 12 + (n – 1) × 2
⇒ 1796 = 12 + 2 n – 2
⇒ 1796 = 12 – 2 + 2 n
⇒ 1796 = 10 + 2 n
After transposing 10 to LHS
⇒ 1796 – 10 = 2 n
⇒ 1786 = 2 n
After rearranging the above expression
⇒ 2 n = 1786
After transposing 2 to RHS
⇒ n = 1786/2
⇒ n = 893
Thus, the number of terms of even numbers from 12 to 1796 = 893
This means 1796 is the 893th term.
Finding the sum of the given even numbers from 12 to 1796
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 1796
= 893/2 (12 + 1796)
= 893/2 × 1808
= 893 × 1808/2
= 1614544/2 = 807272
Thus, the sum of all terms of the given even numbers from 12 to 1796 = 807272
And, the total number of terms = 893
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 1796
= 807272/893 = 904
Thus, the average of the given even numbers from 12 to 1796 = 904 Answer
Similar Questions
(1) Find the average of the first 2142 even numbers.
(2) Find the average of the first 3725 odd numbers.
(3) Find the average of the first 2704 even numbers.
(4) Find the average of the first 2029 odd numbers.
(5) What will be the average of the first 4941 odd numbers?
(6) Find the average of odd numbers from 7 to 959
(7) What will be the average of the first 4318 odd numbers?
(8) What is the average of the first 1333 even numbers?
(9) Find the average of odd numbers from 3 to 1337
(10) Find the average of odd numbers from 3 to 1177