Question : Find the average of even numbers from 4 to 292
Correct Answer 148
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 292
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 292 are
4, 6, 8, . . . . 292
After observing the above list of the even numbers from 4 to 292 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 292 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 292
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 292
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 4 to 292
= 4 + 292/2
= 296/2 = 148
Thus, the average of the even numbers from 4 to 292 = 148 Answer
Method (2) to find the average of the even numbers from 4 to 292
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 292 are
4, 6, 8, . . . . 292
The even numbers from 4 to 292 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 292
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 4 to 292
292 = 4 + (n – 1) × 2
⇒ 292 = 4 + 2 n – 2
⇒ 292 = 4 – 2 + 2 n
⇒ 292 = 2 + 2 n
After transposing 2 to LHS
⇒ 292 – 2 = 2 n
⇒ 290 = 2 n
After rearranging the above expression
⇒ 2 n = 290
After transposing 2 to RHS
⇒ n = 290/2
⇒ n = 145
Thus, the number of terms of even numbers from 4 to 292 = 145
This means 292 is the 145th term.
Finding the sum of the given even numbers from 4 to 292
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 4 to 292
= 145/2 (4 + 292)
= 145/2 × 296
= 145 × 296/2
= 42920/2 = 21460
Thus, the sum of all terms of the given even numbers from 4 to 292 = 21460
And, the total number of terms = 145
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 4 to 292
= 21460/145 = 148
Thus, the average of the given even numbers from 4 to 292 = 148 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 1099
(2) Find the average of the first 3875 even numbers.
(3) Find the average of the first 3454 odd numbers.
(4) What is the average of the first 1583 even numbers?
(5) What is the average of the first 170 odd numbers?
(6) What will be the average of the first 4303 odd numbers?
(7) What is the average of the first 1672 even numbers?
(8) Find the average of the first 3955 odd numbers.
(9) Find the average of odd numbers from 3 to 581
(10) Find the average of the first 3768 even numbers.