Question : Find the average of even numbers from 6 to 1414
Correct Answer 710
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1414
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1414 are
6, 8, 10, . . . . 1414
After observing the above list of the even numbers from 6 to 1414 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1414 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1414
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1414
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 1414
= 6 + 1414/2
= 1420/2 = 710
Thus, the average of the even numbers from 6 to 1414 = 710 Answer
Method (2) to find the average of the even numbers from 6 to 1414
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1414 are
6, 8, 10, . . . . 1414
The even numbers from 6 to 1414 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1414
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 1414
1414 = 6 + (n – 1) × 2
⇒ 1414 = 6 + 2 n – 2
⇒ 1414 = 6 – 2 + 2 n
⇒ 1414 = 4 + 2 n
After transposing 4 to LHS
⇒ 1414 – 4 = 2 n
⇒ 1410 = 2 n
After rearranging the above expression
⇒ 2 n = 1410
After transposing 2 to RHS
⇒ n = 1410/2
⇒ n = 705
Thus, the number of terms of even numbers from 6 to 1414 = 705
This means 1414 is the 705th term.
Finding the sum of the given even numbers from 6 to 1414
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 1414
= 705/2 (6 + 1414)
= 705/2 × 1420
= 705 × 1420/2
= 1001100/2 = 500550
Thus, the sum of all terms of the given even numbers from 6 to 1414 = 500550
And, the total number of terms = 705
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 1414
= 500550/705 = 710
Thus, the average of the given even numbers from 6 to 1414 = 710 Answer
Similar Questions
(1) What will be the average of the first 4841 odd numbers?
(2) What is the average of the first 799 even numbers?
(3) Find the average of odd numbers from 3 to 131
(4) Find the average of the first 4553 even numbers.
(5) Find the average of even numbers from 8 to 1086
(6) Find the average of odd numbers from 9 to 1435
(7) Find the average of the first 3043 odd numbers.
(8) Find the average of the first 3998 odd numbers.
(9) Find the average of the first 2097 odd numbers.
(10) Find the average of odd numbers from 3 to 1377