Question:
Find the average of even numbers from 8 to 1196
Correct Answer
602
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 1196
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 1196 are
8, 10, 12, . . . . 1196
After observing the above list of the even numbers from 8 to 1196 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 1196 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 1196
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1196
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 1196
= 8 + 1196/2
= 1204/2 = 602
Thus, the average of the even numbers from 8 to 1196 = 602 Answer
Method (2) to find the average of the even numbers from 8 to 1196
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 1196 are
8, 10, 12, . . . . 1196
The even numbers from 8 to 1196 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1196
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 1196
1196 = 8 + (n – 1) × 2
⇒ 1196 = 8 + 2 n – 2
⇒ 1196 = 8 – 2 + 2 n
⇒ 1196 = 6 + 2 n
After transposing 6 to LHS
⇒ 1196 – 6 = 2 n
⇒ 1190 = 2 n
After rearranging the above expression
⇒ 2 n = 1190
After transposing 2 to RHS
⇒ n = 1190/2
⇒ n = 595
Thus, the number of terms of even numbers from 8 to 1196 = 595
This means 1196 is the 595th term.
Finding the sum of the given even numbers from 8 to 1196
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 1196
= 595/2 (8 + 1196)
= 595/2 × 1204
= 595 × 1204/2
= 716380/2 = 358190
Thus, the sum of all terms of the given even numbers from 8 to 1196 = 358190
And, the total number of terms = 595
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 1196
= 358190/595 = 602
Thus, the average of the given even numbers from 8 to 1196 = 602 Answer
Similar Questions
(1) Find the average of the first 923 odd numbers.
(2) Find the average of odd numbers from 11 to 281
(3) Find the average of odd numbers from 5 to 1357
(4) Find the average of odd numbers from 15 to 1287
(5) Find the average of the first 1506 odd numbers.
(6) What is the average of the first 1340 even numbers?
(7) Find the average of odd numbers from 5 to 993
(8) Find the average of even numbers from 4 to 1252
(9) Find the average of odd numbers from 7 to 493
(10) Find the average of the first 3259 odd numbers.