Question : Find the average of even numbers from 10 to 1980
Correct Answer 995
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1980
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1980 are
10, 12, 14, . . . . 1980
After observing the above list of the even numbers from 10 to 1980 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1980 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1980
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1980
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 1980
= 10 + 1980/2
= 1990/2 = 995
Thus, the average of the even numbers from 10 to 1980 = 995 Answer
Method (2) to find the average of the even numbers from 10 to 1980
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1980 are
10, 12, 14, . . . . 1980
The even numbers from 10 to 1980 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1980
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 1980
1980 = 10 + (n – 1) × 2
⇒ 1980 = 10 + 2 n – 2
⇒ 1980 = 10 – 2 + 2 n
⇒ 1980 = 8 + 2 n
After transposing 8 to LHS
⇒ 1980 – 8 = 2 n
⇒ 1972 = 2 n
After rearranging the above expression
⇒ 2 n = 1972
After transposing 2 to RHS
⇒ n = 1972/2
⇒ n = 986
Thus, the number of terms of even numbers from 10 to 1980 = 986
This means 1980 is the 986th term.
Finding the sum of the given even numbers from 10 to 1980
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 1980
= 986/2 (10 + 1980)
= 986/2 × 1990
= 986 × 1990/2
= 1962140/2 = 981070
Thus, the sum of all terms of the given even numbers from 10 to 1980 = 981070
And, the total number of terms = 986
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 1980
= 981070/986 = 995
Thus, the average of the given even numbers from 10 to 1980 = 995 Answer
Similar Questions
(1) Find the average of the first 861 odd numbers.
(2) Find the average of the first 2446 even numbers.
(3) Find the average of odd numbers from 11 to 193
(4) Find the average of even numbers from 6 to 774
(5) Find the average of the first 3912 odd numbers.
(6) Find the average of the first 1287 odd numbers.
(7) Find the average of the first 4870 even numbers.
(8) Find the average of even numbers from 6 to 1116
(9) Find the average of odd numbers from 3 to 1389
(10) Find the average of the first 3079 even numbers.