Question:
Find the average of even numbers from 6 to 318
Correct Answer
162
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 318
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 318 are
6, 8, 10, . . . . 318
After observing the above list of the even numbers from 6 to 318 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 318 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 318
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 318
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 318
= 6 + 318/2
= 324/2 = 162
Thus, the average of the even numbers from 6 to 318 = 162 Answer
Method (2) to find the average of the even numbers from 6 to 318
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 318 are
6, 8, 10, . . . . 318
The even numbers from 6 to 318 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 318
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 318
318 = 6 + (n – 1) × 2
⇒ 318 = 6 + 2 n – 2
⇒ 318 = 6 – 2 + 2 n
⇒ 318 = 4 + 2 n
After transposing 4 to LHS
⇒ 318 – 4 = 2 n
⇒ 314 = 2 n
After rearranging the above expression
⇒ 2 n = 314
After transposing 2 to RHS
⇒ n = 314/2
⇒ n = 157
Thus, the number of terms of even numbers from 6 to 318 = 157
This means 318 is the 157th term.
Finding the sum of the given even numbers from 6 to 318
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 318
= 157/2 (6 + 318)
= 157/2 × 324
= 157 × 324/2
= 50868/2 = 25434
Thus, the sum of all terms of the given even numbers from 6 to 318 = 25434
And, the total number of terms = 157
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 318
= 25434/157 = 162
Thus, the average of the given even numbers from 6 to 318 = 162 Answer
Similar Questions
(1) What is the average of the first 1173 even numbers?
(2) Find the average of the first 3908 odd numbers.
(3) What will be the average of the first 4987 odd numbers?
(4) Find the average of even numbers from 4 to 1400
(5) Find the average of the first 2732 even numbers.
(6) Find the average of even numbers from 4 to 1278
(7) Find the average of even numbers from 12 to 534
(8) Find the average of the first 1386 odd numbers.
(9) Find the average of odd numbers from 13 to 999
(10) Find the average of the first 3555 even numbers.