Question : Find the average of even numbers from 6 to 1118
Correct Answer 562
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1118
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1118 are
6, 8, 10, . . . . 1118
After observing the above list of the even numbers from 6 to 1118 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1118 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1118
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1118
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 1118
= 6 + 1118/2
= 1124/2 = 562
Thus, the average of the even numbers from 6 to 1118 = 562 Answer
Method (2) to find the average of the even numbers from 6 to 1118
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1118 are
6, 8, 10, . . . . 1118
The even numbers from 6 to 1118 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1118
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 1118
1118 = 6 + (n – 1) × 2
⇒ 1118 = 6 + 2 n – 2
⇒ 1118 = 6 – 2 + 2 n
⇒ 1118 = 4 + 2 n
After transposing 4 to LHS
⇒ 1118 – 4 = 2 n
⇒ 1114 = 2 n
After rearranging the above expression
⇒ 2 n = 1114
After transposing 2 to RHS
⇒ n = 1114/2
⇒ n = 557
Thus, the number of terms of even numbers from 6 to 1118 = 557
This means 1118 is the 557th term.
Finding the sum of the given even numbers from 6 to 1118
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 1118
= 557/2 (6 + 1118)
= 557/2 × 1124
= 557 × 1124/2
= 626068/2 = 313034
Thus, the sum of all terms of the given even numbers from 6 to 1118 = 313034
And, the total number of terms = 557
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 1118
= 313034/557 = 562
Thus, the average of the given even numbers from 6 to 1118 = 562 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 264
(2) Find the average of the first 3927 odd numbers.
(3) Find the average of odd numbers from 15 to 1713
(4) Find the average of the first 2393 odd numbers.
(5) What is the average of the first 485 even numbers?
(6) Find the average of the first 4289 even numbers.
(7) Find the average of the first 4722 even numbers.
(8) Find the average of odd numbers from 5 to 267
(9) Find the average of even numbers from 4 to 92
(10) Find the average of even numbers from 12 to 100