Question:
Find the average of even numbers from 8 to 1434
Correct Answer
721
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 1434
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 1434 are
8, 10, 12, . . . . 1434
After observing the above list of the even numbers from 8 to 1434 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 1434 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 1434
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1434
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 1434
= 8 + 1434/2
= 1442/2 = 721
Thus, the average of the even numbers from 8 to 1434 = 721 Answer
Method (2) to find the average of the even numbers from 8 to 1434
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 1434 are
8, 10, 12, . . . . 1434
The even numbers from 8 to 1434 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1434
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 1434
1434 = 8 + (n – 1) × 2
⇒ 1434 = 8 + 2 n – 2
⇒ 1434 = 8 – 2 + 2 n
⇒ 1434 = 6 + 2 n
After transposing 6 to LHS
⇒ 1434 – 6 = 2 n
⇒ 1428 = 2 n
After rearranging the above expression
⇒ 2 n = 1428
After transposing 2 to RHS
⇒ n = 1428/2
⇒ n = 714
Thus, the number of terms of even numbers from 8 to 1434 = 714
This means 1434 is the 714th term.
Finding the sum of the given even numbers from 8 to 1434
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 1434
= 714/2 (8 + 1434)
= 714/2 × 1442
= 714 × 1442/2
= 1029588/2 = 514794
Thus, the sum of all terms of the given even numbers from 8 to 1434 = 514794
And, the total number of terms = 714
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 1434
= 514794/714 = 721
Thus, the average of the given even numbers from 8 to 1434 = 721 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 1986
(2) Find the average of the first 659 odd numbers.
(3) Find the average of the first 1128 odd numbers.
(4) Find the average of even numbers from 12 to 762
(5) Find the average of odd numbers from 11 to 1241
(6) What will be the average of the first 4389 odd numbers?
(7) Find the average of even numbers from 12 to 908
(8) Find the average of even numbers from 4 to 1856
(9) Find the average of even numbers from 8 to 442
(10) What is the average of the first 1510 even numbers?