Question:
Find the average of even numbers from 10 to 1994
Correct Answer
1002
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1994
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1994 are
10, 12, 14, . . . . 1994
After observing the above list of the even numbers from 10 to 1994 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1994 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1994
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1994
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 10 to 1994
= 10 + 1994/2
= 2004/2 = 1002
Thus, the average of the even numbers from 10 to 1994 = 1002 Answer
Method (2) to find the average of the even numbers from 10 to 1994
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1994 are
10, 12, 14, . . . . 1994
The even numbers from 10 to 1994 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1994
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 10 to 1994
1994 = 10 + (n – 1) × 2
⇒ 1994 = 10 + 2 n – 2
⇒ 1994 = 10 – 2 + 2 n
⇒ 1994 = 8 + 2 n
After transposing 8 to LHS
⇒ 1994 – 8 = 2 n
⇒ 1986 = 2 n
After rearranging the above expression
⇒ 2 n = 1986
After transposing 2 to RHS
⇒ n = 1986/2
⇒ n = 993
Thus, the number of terms of even numbers from 10 to 1994 = 993
This means 1994 is the 993th term.
Finding the sum of the given even numbers from 10 to 1994
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 10 to 1994
= 993/2 (10 + 1994)
= 993/2 × 2004
= 993 × 2004/2
= 1989972/2 = 994986
Thus, the sum of all terms of the given even numbers from 10 to 1994 = 994986
And, the total number of terms = 993
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 10 to 1994
= 994986/993 = 1002
Thus, the average of the given even numbers from 10 to 1994 = 1002 Answer
Similar Questions
(1) Find the average of the first 3548 even numbers.
(2) Find the average of odd numbers from 11 to 579
(3) Find the average of the first 1425 odd numbers.
(4) Find the average of odd numbers from 13 to 1331
(5) Find the average of the first 4503 even numbers.
(6) Find the average of the first 3980 odd numbers.
(7) Find the average of odd numbers from 5 to 895
(8) Find the average of odd numbers from 5 to 569
(9) Find the average of odd numbers from 3 to 1007
(10) Find the average of the first 3040 even numbers.