Question : Find the average of even numbers from 8 to 1176
Correct Answer 592
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 1176
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 1176 are
8, 10, 12, . . . . 1176
After observing the above list of the even numbers from 8 to 1176 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 1176 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 1176
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1176
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 8 to 1176
= 8 + 1176/2
= 1184/2 = 592
Thus, the average of the even numbers from 8 to 1176 = 592 Answer
Method (2) to find the average of the even numbers from 8 to 1176
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 1176 are
8, 10, 12, . . . . 1176
The even numbers from 8 to 1176 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1176
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 8 to 1176
1176 = 8 + (n – 1) × 2
⇒ 1176 = 8 + 2 n – 2
⇒ 1176 = 8 – 2 + 2 n
⇒ 1176 = 6 + 2 n
After transposing 6 to LHS
⇒ 1176 – 6 = 2 n
⇒ 1170 = 2 n
After rearranging the above expression
⇒ 2 n = 1170
After transposing 2 to RHS
⇒ n = 1170/2
⇒ n = 585
Thus, the number of terms of even numbers from 8 to 1176 = 585
This means 1176 is the 585th term.
Finding the sum of the given even numbers from 8 to 1176
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 8 to 1176
= 585/2 (8 + 1176)
= 585/2 × 1184
= 585 × 1184/2
= 692640/2 = 346320
Thus, the sum of all terms of the given even numbers from 8 to 1176 = 346320
And, the total number of terms = 585
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 8 to 1176
= 346320/585 = 592
Thus, the average of the given even numbers from 8 to 1176 = 592 Answer
Similar Questions
(1) What is the average of the first 214 even numbers?
(2) Find the average of odd numbers from 13 to 305
(3) Find the average of the first 1461 odd numbers.
(4) What is the average of the first 193 even numbers?
(5) Find the average of even numbers from 12 to 812
(6) What is the average of the first 915 even numbers?
(7) Find the average of even numbers from 6 to 698
(8) Find the average of the first 638 odd numbers.
(9) Find the average of the first 2287 even numbers.
(10) Find the average of the first 4824 even numbers.