Question : Find the average of even numbers from 8 to 1130
Correct Answer 569
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 1130
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 1130 are
8, 10, 12, . . . . 1130
After observing the above list of the even numbers from 8 to 1130 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 1130 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 1130
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1130
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 1130
= 8 + 1130/2
= 1138/2 = 569
Thus, the average of the even numbers from 8 to 1130 = 569 Answer
Method (2) to find the average of the even numbers from 8 to 1130
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 1130 are
8, 10, 12, . . . . 1130
The even numbers from 8 to 1130 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1130
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 1130
1130 = 8 + (n – 1) × 2
⇒ 1130 = 8 + 2 n – 2
⇒ 1130 = 8 – 2 + 2 n
⇒ 1130 = 6 + 2 n
After transposing 6 to LHS
⇒ 1130 – 6 = 2 n
⇒ 1124 = 2 n
After rearranging the above expression
⇒ 2 n = 1124
After transposing 2 to RHS
⇒ n = 1124/2
⇒ n = 562
Thus, the number of terms of even numbers from 8 to 1130 = 562
This means 1130 is the 562th term.
Finding the sum of the given even numbers from 8 to 1130
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 1130
= 562/2 (8 + 1130)
= 562/2 × 1138
= 562 × 1138/2
= 639556/2 = 319778
Thus, the sum of all terms of the given even numbers from 8 to 1130 = 319778
And, the total number of terms = 562
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 1130
= 319778/562 = 569
Thus, the average of the given even numbers from 8 to 1130 = 569 Answer
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