Question : Find the average of even numbers from 6 to 308
Correct Answer 157
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 308
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 308 are
6, 8, 10, . . . . 308
After observing the above list of the even numbers from 6 to 308 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 308 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 308
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 308
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 308
= 6 + 308/2
= 314/2 = 157
Thus, the average of the even numbers from 6 to 308 = 157 Answer
Method (2) to find the average of the even numbers from 6 to 308
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 308 are
6, 8, 10, . . . . 308
The even numbers from 6 to 308 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 308
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 308
308 = 6 + (n – 1) × 2
⇒ 308 = 6 + 2 n – 2
⇒ 308 = 6 – 2 + 2 n
⇒ 308 = 4 + 2 n
After transposing 4 to LHS
⇒ 308 – 4 = 2 n
⇒ 304 = 2 n
After rearranging the above expression
⇒ 2 n = 304
After transposing 2 to RHS
⇒ n = 304/2
⇒ n = 152
Thus, the number of terms of even numbers from 6 to 308 = 152
This means 308 is the 152th term.
Finding the sum of the given even numbers from 6 to 308
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 308
= 152/2 (6 + 308)
= 152/2 × 314
= 152 × 314/2
= 47728/2 = 23864
Thus, the sum of all terms of the given even numbers from 6 to 308 = 23864
And, the total number of terms = 152
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 308
= 23864/152 = 157
Thus, the average of the given even numbers from 6 to 308 = 157 Answer
Similar Questions
(1) Find the average of the first 4830 even numbers.
(2) What will be the average of the first 4683 odd numbers?
(3) Find the average of even numbers from 12 to 1468
(4) Find the average of odd numbers from 3 to 173
(5) Find the average of odd numbers from 9 to 903
(6) Find the average of the first 3328 odd numbers.
(7) Find the average of even numbers from 10 to 742
(8) Find the average of odd numbers from 3 to 43
(9) Find the average of even numbers from 4 to 1560
(10) Find the average of even numbers from 10 to 378