Question:
Find the average of even numbers from 8 to 1310
Correct Answer
659
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 1310
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 1310 are
8, 10, 12, . . . . 1310
After observing the above list of the even numbers from 8 to 1310 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 1310 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 1310
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1310
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 1310
= 8 + 1310/2
= 1318/2 = 659
Thus, the average of the even numbers from 8 to 1310 = 659 Answer
Method (2) to find the average of the even numbers from 8 to 1310
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 1310 are
8, 10, 12, . . . . 1310
The even numbers from 8 to 1310 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1310
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 1310
1310 = 8 + (n – 1) × 2
⇒ 1310 = 8 + 2 n – 2
⇒ 1310 = 8 – 2 + 2 n
⇒ 1310 = 6 + 2 n
After transposing 6 to LHS
⇒ 1310 – 6 = 2 n
⇒ 1304 = 2 n
After rearranging the above expression
⇒ 2 n = 1304
After transposing 2 to RHS
⇒ n = 1304/2
⇒ n = 652
Thus, the number of terms of even numbers from 8 to 1310 = 652
This means 1310 is the 652th term.
Finding the sum of the given even numbers from 8 to 1310
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 1310
= 652/2 (8 + 1310)
= 652/2 × 1318
= 652 × 1318/2
= 859336/2 = 429668
Thus, the sum of all terms of the given even numbers from 8 to 1310 = 429668
And, the total number of terms = 652
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 1310
= 429668/652 = 659
Thus, the average of the given even numbers from 8 to 1310 = 659 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 968
(2) Find the average of even numbers from 4 to 918
(3) Find the average of the first 953 odd numbers.
(4) Find the average of even numbers from 12 to 640
(5) Find the average of the first 2080 even numbers.
(6) What is the average of the first 239 even numbers?
(7) Find the average of even numbers from 10 to 1802
(8) Find the average of odd numbers from 3 to 323
(9) What is the average of the first 409 even numbers?
(10) Find the average of the first 2718 even numbers.