Question : Find the average of even numbers from 10 to 1130
Correct Answer 570
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1130
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1130 are
10, 12, 14, . . . . 1130
After observing the above list of the even numbers from 10 to 1130 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1130 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1130
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1130
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 1130
= 10 + 1130/2
= 1140/2 = 570
Thus, the average of the even numbers from 10 to 1130 = 570 Answer
Method (2) to find the average of the even numbers from 10 to 1130
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1130 are
10, 12, 14, . . . . 1130
The even numbers from 10 to 1130 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1130
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 1130
1130 = 10 + (n – 1) × 2
⇒ 1130 = 10 + 2 n – 2
⇒ 1130 = 10 – 2 + 2 n
⇒ 1130 = 8 + 2 n
After transposing 8 to LHS
⇒ 1130 – 8 = 2 n
⇒ 1122 = 2 n
After rearranging the above expression
⇒ 2 n = 1122
After transposing 2 to RHS
⇒ n = 1122/2
⇒ n = 561
Thus, the number of terms of even numbers from 10 to 1130 = 561
This means 1130 is the 561th term.
Finding the sum of the given even numbers from 10 to 1130
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 1130
= 561/2 (10 + 1130)
= 561/2 × 1140
= 561 × 1140/2
= 639540/2 = 319770
Thus, the sum of all terms of the given even numbers from 10 to 1130 = 319770
And, the total number of terms = 561
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 1130
= 319770/561 = 570
Thus, the average of the given even numbers from 10 to 1130 = 570 Answer
Similar Questions
(1) What is the average of the first 407 even numbers?
(2) Find the average of odd numbers from 5 to 611
(3) What is the average of the first 449 even numbers?
(4) Find the average of even numbers from 4 to 1394
(5) Find the average of odd numbers from 11 to 839
(6) Find the average of odd numbers from 5 to 1411
(7) What will be the average of the first 5000 odd numbers?
(8) Find the average of odd numbers from 9 to 599
(9) Find the average of even numbers from 6 to 1304
(10) What is the average of the first 341 even numbers?