Question : Find the average of odd numbers from 5 to 217
Correct Answer 111
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 217
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 217 are
5, 7, 9, . . . . 217
After observing the above list of the odd numbers from 5 to 217 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 217 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 217
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 217
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 odd numbers from 5 to 217
= 5 + 217/2
= 222/2 = 111
Thus, the average of the odd numbers from 5 to 217 = 111 Answer
Method (2) to find the average of the odd numbers from 5 to 217
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 217 are
5, 7, 9, . . . . 217
The odd numbers from 5 to 217 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 217
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 odd numbers from 5 to 217
217 = 5 + (n – 1) × 2
⇒ 217 = 5 + 2 n – 2
⇒ 217 = 5 – 2 + 2 n
⇒ 217 = 3 + 2 n
After transposing 3 to LHS
⇒ 217 – 3 = 2 n
⇒ 214 = 2 n
After rearranging the above expression
⇒ 2 n = 214
After transposing 2 to RHS
⇒ n = 214/2
⇒ n = 107
Thus, the number of terms of odd numbers from 5 to 217 = 107
This means 217 is the 107th term.
Finding the sum of the given odd numbers from 5 to 217
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 odd numbers from 5 to 217
= 107/2 (5 + 217)
= 107/2 × 222
= 107 × 222/2
= 23754/2 = 11877
Thus, the sum of all terms of the given odd numbers from 5 to 217 = 11877
And, the total number of terms = 107
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given odd numbers from 5 to 217
= 11877/107 = 111
Thus, the average of the given odd numbers from 5 to 217 = 111 Answer
Similar Questions
(1) What is the average of the first 1617 even numbers?
(2) Find the average of the first 4546 even numbers.
(3) Find the average of odd numbers from 7 to 1075
(4) Find the average of even numbers from 4 to 1500
(5) Find the average of the first 3528 odd numbers.
(6) Find the average of the first 849 odd numbers.
(7) Find the average of the first 1660 odd numbers.
(8) Find the average of the first 3082 even numbers.
(9) What is the average of the first 1669 even numbers?
(10) Find the average of even numbers from 8 to 530