Question:
Find the average of even numbers from 8 to 514
Correct Answer
261
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 514
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 514 are
8, 10, 12, . . . . 514
After observing the above list of the even numbers from 8 to 514 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 514 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 514
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 514
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 8 to 514
= 8 + 514/2
= 522/2 = 261
Thus, the average of the even numbers from 8 to 514 = 261 Answer
Method (2) to find the average of the even numbers from 8 to 514
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 514 are
8, 10, 12, . . . . 514
The even numbers from 8 to 514 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 514
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 8 to 514
514 = 8 + (n – 1) × 2
⇒ 514 = 8 + 2 n – 2
⇒ 514 = 8 – 2 + 2 n
⇒ 514 = 6 + 2 n
After transposing 6 to LHS
⇒ 514 – 6 = 2 n
⇒ 508 = 2 n
After rearranging the above expression
⇒ 2 n = 508
After transposing 2 to RHS
⇒ n = 508/2
⇒ n = 254
Thus, the number of terms of even numbers from 8 to 514 = 254
This means 514 is the 254th term.
Finding the sum of the given even numbers from 8 to 514
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 8 to 514
= 254/2 (8 + 514)
= 254/2 × 522
= 254 × 522/2
= 132588/2 = 66294
Thus, the sum of all terms of the given even numbers from 8 to 514 = 66294
And, the total number of terms = 254
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 8 to 514
= 66294/254 = 261
Thus, the average of the given even numbers from 8 to 514 = 261 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 979
(2) What is the average of the first 1797 even numbers?
(3) Find the average of odd numbers from 3 to 991
(4) Find the average of the first 3496 odd numbers.
(5) Find the average of the first 3808 even numbers.
(6) Find the average of even numbers from 10 to 614
(7) Find the average of the first 3431 even numbers.
(8) Find the average of even numbers from 12 to 1818
(9) What will be the average of the first 4933 odd numbers?
(10) Find the average of odd numbers from 15 to 1455