Question:
Find the average of odd numbers from 15 to 207
Correct Answer
111
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 207
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 207 are
15, 17, 19, . . . . 207
After observing the above list of the odd numbers from 15 to 207 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 207 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 207
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 207
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 15 to 207
= 15 + 207/2
= 222/2 = 111
Thus, the average of the odd numbers from 15 to 207 = 111 Answer
Method (2) to find the average of the odd numbers from 15 to 207
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 207 are
15, 17, 19, . . . . 207
The odd numbers from 15 to 207 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 207
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 15 to 207
207 = 15 + (n – 1) × 2
⇒ 207 = 15 + 2 n – 2
⇒ 207 = 15 – 2 + 2 n
⇒ 207 = 13 + 2 n
After transposing 13 to LHS
⇒ 207 – 13 = 2 n
⇒ 194 = 2 n
After rearranging the above expression
⇒ 2 n = 194
After transposing 2 to RHS
⇒ n = 194/2
⇒ n = 97
Thus, the number of terms of odd numbers from 15 to 207 = 97
This means 207 is the 97th term.
Finding the sum of the given odd numbers from 15 to 207
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 15 to 207
= 97/2 (15 + 207)
= 97/2 × 222
= 97 × 222/2
= 21534/2 = 10767
Thus, the sum of all terms of the given odd numbers from 15 to 207 = 10767
And, the total number of terms = 97
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 15 to 207
= 10767/97 = 111
Thus, the average of the given odd numbers from 15 to 207 = 111 Answer
Similar Questions
(1) Find the average of the first 1888 odd numbers.
(2) Find the average of odd numbers from 11 to 233
(3) Find the average of the first 2831 even numbers.
(4) Find the average of the first 1860 odd numbers.
(5) Find the average of the first 211 odd numbers.
(6) What is the average of the first 517 even numbers?
(7) What will be the average of the first 4056 odd numbers?
(8) Find the average of even numbers from 10 to 1696
(9) Find the average of even numbers from 8 to 492
(10) Find the average of the first 2166 even numbers.