Question:
Find the average of even numbers from 10 to 266
Correct Answer
138
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 266
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 266 are
10, 12, 14, . . . . 266
After observing the above list of the even numbers from 10 to 266 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 266 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 266
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 266
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 10 to 266
= 10 + 266/2
= 276/2 = 138
Thus, the average of the even numbers from 10 to 266 = 138 Answer
Method (2) to find the average of the even numbers from 10 to 266
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 266 are
10, 12, 14, . . . . 266
The even numbers from 10 to 266 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 266
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 10 to 266
266 = 10 + (n – 1) × 2
⇒ 266 = 10 + 2 n – 2
⇒ 266 = 10 – 2 + 2 n
⇒ 266 = 8 + 2 n
After transposing 8 to LHS
⇒ 266 – 8 = 2 n
⇒ 258 = 2 n
After rearranging the above expression
⇒ 2 n = 258
After transposing 2 to RHS
⇒ n = 258/2
⇒ n = 129
Thus, the number of terms of even numbers from 10 to 266 = 129
This means 266 is the 129th term.
Finding the sum of the given even numbers from 10 to 266
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 10 to 266
= 129/2 (10 + 266)
= 129/2 × 276
= 129 × 276/2
= 35604/2 = 17802
Thus, the sum of all terms of the given even numbers from 10 to 266 = 17802
And, the total number of terms = 129
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 10 to 266
= 17802/129 = 138
Thus, the average of the given even numbers from 10 to 266 = 138 Answer
Similar Questions
(1) Find the average of the first 2067 even numbers.
(2) Find the average of the first 2838 odd numbers.
(3) Find the average of even numbers from 6 to 1398
(4) Find the average of even numbers from 12 to 1444
(5) What is the average of the first 1911 even numbers?
(6) Find the average of the first 1770 odd numbers.
(7) Find the average of even numbers from 10 to 1676
(8) Find the average of the first 3942 even numbers.
(9) Find the average of the first 2923 even numbers.
(10) Find the average of the first 2601 even numbers.