Question:
Find the average of even numbers from 6 to 862
Correct Answer
434
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 862
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 862 are
6, 8, 10, . . . . 862
After observing the above list of the even numbers from 6 to 862 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 862 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 862
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 862
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 6 to 862
= 6 + 862/2
= 868/2 = 434
Thus, the average of the even numbers from 6 to 862 = 434 Answer
Method (2) to find the average of the even numbers from 6 to 862
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 862 are
6, 8, 10, . . . . 862
The even numbers from 6 to 862 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 862
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 6 to 862
862 = 6 + (n – 1) × 2
⇒ 862 = 6 + 2 n – 2
⇒ 862 = 6 – 2 + 2 n
⇒ 862 = 4 + 2 n
After transposing 4 to LHS
⇒ 862 – 4 = 2 n
⇒ 858 = 2 n
After rearranging the above expression
⇒ 2 n = 858
After transposing 2 to RHS
⇒ n = 858/2
⇒ n = 429
Thus, the number of terms of even numbers from 6 to 862 = 429
This means 862 is the 429th term.
Finding the sum of the given even numbers from 6 to 862
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 6 to 862
= 429/2 (6 + 862)
= 429/2 × 868
= 429 × 868/2
= 372372/2 = 186186
Thus, the sum of all terms of the given even numbers from 6 to 862 = 186186
And, the total number of terms = 429
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 6 to 862
= 186186/429 = 434
Thus, the average of the given even numbers from 6 to 862 = 434 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 410
(2) Find the average of the first 331 odd numbers.
(3) Find the average of odd numbers from 11 to 1471
(4) Find the average of even numbers from 10 to 202
(5) Find the average of even numbers from 12 to 1480
(6) Find the average of the first 2794 odd numbers.
(7) What is the average of the first 1606 even numbers?
(8) Find the average of odd numbers from 3 to 1263
(9) What will be the average of the first 4105 odd numbers?
(10) Find the average of even numbers from 6 to 1300