Question:
Find the average of even numbers from 12 to 510
Correct Answer
261
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 510
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 510 are
12, 14, 16, . . . . 510
After observing the above list of the even numbers from 12 to 510 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 510 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 510
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 510
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 510
= 12 + 510/2
= 522/2 = 261
Thus, the average of the even numbers from 12 to 510 = 261 Answer
Method (2) to find the average of the even numbers from 12 to 510
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 510 are
12, 14, 16, . . . . 510
The even numbers from 12 to 510 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 510
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 510
510 = 12 + (n – 1) × 2
⇒ 510 = 12 + 2 n – 2
⇒ 510 = 12 – 2 + 2 n
⇒ 510 = 10 + 2 n
After transposing 10 to LHS
⇒ 510 – 10 = 2 n
⇒ 500 = 2 n
After rearranging the above expression
⇒ 2 n = 500
After transposing 2 to RHS
⇒ n = 500/2
⇒ n = 250
Thus, the number of terms of even numbers from 12 to 510 = 250
This means 510 is the 250th term.
Finding the sum of the given even numbers from 12 to 510
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 510
= 250/2 (12 + 510)
= 250/2 × 522
= 250 × 522/2
= 130500/2 = 65250
Thus, the sum of all terms of the given even numbers from 12 to 510 = 65250
And, the total number of terms = 250
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 510
= 65250/250 = 261
Thus, the average of the given even numbers from 12 to 510 = 261 Answer
Similar Questions
(1) Find the average of odd numbers from 5 to 329
(2) Find the average of odd numbers from 15 to 1315
(3) Find the average of odd numbers from 3 to 723
(4) Find the average of the first 1322 odd numbers.
(5) Find the average of the first 4289 even numbers.
(6) Find the average of even numbers from 10 to 1606
(7) Find the average of odd numbers from 13 to 1479
(8) What will be the average of the first 4263 odd numbers?
(9) Find the average of odd numbers from 15 to 1597
(10) Find the average of odd numbers from 5 to 1365