Question : Find the average of even numbers from 6 to 332
Correct Answer 169
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 332
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 332 are
6, 8, 10, . . . . 332
After observing the above list of the even numbers from 6 to 332 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 332 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 332
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 332
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 332
= 6 + 332/2
= 338/2 = 169
Thus, the average of the even numbers from 6 to 332 = 169 Answer
Method (2) to find the average of the even numbers from 6 to 332
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 332 are
6, 8, 10, . . . . 332
The even numbers from 6 to 332 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 332
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 332
332 = 6 + (n – 1) × 2
⇒ 332 = 6 + 2 n – 2
⇒ 332 = 6 – 2 + 2 n
⇒ 332 = 4 + 2 n
After transposing 4 to LHS
⇒ 332 – 4 = 2 n
⇒ 328 = 2 n
After rearranging the above expression
⇒ 2 n = 328
After transposing 2 to RHS
⇒ n = 328/2
⇒ n = 164
Thus, the number of terms of even numbers from 6 to 332 = 164
This means 332 is the 164th term.
Finding the sum of the given even numbers from 6 to 332
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 332
= 164/2 (6 + 332)
= 164/2 × 338
= 164 × 338/2
= 55432/2 = 27716
Thus, the sum of all terms of the given even numbers from 6 to 332 = 27716
And, the total number of terms = 164
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 332
= 27716/164 = 169
Thus, the average of the given even numbers from 6 to 332 = 169 Answer
Similar Questions
(1) Find the average of the first 2238 odd numbers.
(2) Find the average of the first 2686 odd numbers.
(3) Find the average of the first 3117 even numbers.
(4) Find the average of odd numbers from 11 to 841
(5) What is the average of the first 562 even numbers?
(6) Find the average of the first 4180 even numbers.
(7) Find the average of the first 794 odd numbers.
(8) Find the average of even numbers from 12 to 886
(9) Find the average of odd numbers from 15 to 1731
(10) Find the average of the first 1212 odd numbers.