Question:
Find the average of even numbers from 8 to 284
Correct Answer
146
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 284
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 284 are
8, 10, 12, . . . . 284
After observing the above list of the even numbers from 8 to 284 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 284 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 284
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 284
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 284
= 8 + 284/2
= 292/2 = 146
Thus, the average of the even numbers from 8 to 284 = 146 Answer
Method (2) to find the average of the even numbers from 8 to 284
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 284 are
8, 10, 12, . . . . 284
The even numbers from 8 to 284 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 284
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 284
284 = 8 + (n – 1) × 2
⇒ 284 = 8 + 2 n – 2
⇒ 284 = 8 – 2 + 2 n
⇒ 284 = 6 + 2 n
After transposing 6 to LHS
⇒ 284 – 6 = 2 n
⇒ 278 = 2 n
After rearranging the above expression
⇒ 2 n = 278
After transposing 2 to RHS
⇒ n = 278/2
⇒ n = 139
Thus, the number of terms of even numbers from 8 to 284 = 139
This means 284 is the 139th term.
Finding the sum of the given even numbers from 8 to 284
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 284
= 139/2 (8 + 284)
= 139/2 × 292
= 139 × 292/2
= 40588/2 = 20294
Thus, the sum of all terms of the given even numbers from 8 to 284 = 20294
And, the total number of terms = 139
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 284
= 20294/139 = 146
Thus, the average of the given even numbers from 8 to 284 = 146 Answer
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