Question : Find the average of even numbers from 6 to 1542
Correct Answer 774
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1542
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1542 are
6, 8, 10, . . . . 1542
After observing the above list of the even numbers from 6 to 1542 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1542 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1542
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1542
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 1542
= 6 + 1542/2
= 1548/2 = 774
Thus, the average of the even numbers from 6 to 1542 = 774 Answer
Method (2) to find the average of the even numbers from 6 to 1542
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1542 are
6, 8, 10, . . . . 1542
The even numbers from 6 to 1542 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1542
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 1542
1542 = 6 + (n – 1) × 2
⇒ 1542 = 6 + 2 n – 2
⇒ 1542 = 6 – 2 + 2 n
⇒ 1542 = 4 + 2 n
After transposing 4 to LHS
⇒ 1542 – 4 = 2 n
⇒ 1538 = 2 n
After rearranging the above expression
⇒ 2 n = 1538
After transposing 2 to RHS
⇒ n = 1538/2
⇒ n = 769
Thus, the number of terms of even numbers from 6 to 1542 = 769
This means 1542 is the 769th term.
Finding the sum of the given even numbers from 6 to 1542
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 1542
= 769/2 (6 + 1542)
= 769/2 × 1548
= 769 × 1548/2
= 1190412/2 = 595206
Thus, the sum of all terms of the given even numbers from 6 to 1542 = 595206
And, the total number of terms = 769
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 1542
= 595206/769 = 774
Thus, the average of the given even numbers from 6 to 1542 = 774 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 340
(2) Find the average of even numbers from 4 to 1744
(3) Find the average of the first 2765 even numbers.
(4) Find the average of even numbers from 12 to 1342
(5) Find the average of odd numbers from 5 to 657
(6) Find the average of the first 553 odd numbers.
(7) Find the average of odd numbers from 7 to 345
(8) Find the average of even numbers from 10 to 1378
(9) Find the average of even numbers from 8 to 282
(10) What is the average of the first 1286 even numbers?