Question:
Find the average of odd numbers from 13 to 289
Correct Answer
151
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 289
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 289 are
13, 15, 17, . . . . 289
After observing the above list of the odd numbers from 13 to 289 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 289 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 289
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 289
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 13 to 289
= 13 + 289/2
= 302/2 = 151
Thus, the average of the odd numbers from 13 to 289 = 151 Answer
Method (2) to find the average of the odd numbers from 13 to 289
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 289 are
13, 15, 17, . . . . 289
The odd numbers from 13 to 289 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 289
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 13 to 289
289 = 13 + (n – 1) × 2
⇒ 289 = 13 + 2 n – 2
⇒ 289 = 13 – 2 + 2 n
⇒ 289 = 11 + 2 n
After transposing 11 to LHS
⇒ 289 – 11 = 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 odd numbers from 13 to 289 = 139
This means 289 is the 139th term.
Finding the sum of the given odd numbers from 13 to 289
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 13 to 289
= 139/2 (13 + 289)
= 139/2 × 302
= 139 × 302/2
= 41978/2 = 20989
Thus, the sum of all terms of the given odd numbers from 13 to 289 = 20989
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 odd numbers from 13 to 289
= 20989/139 = 151
Thus, the average of the given odd numbers from 13 to 289 = 151 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 309
(2) Find the average of odd numbers from 13 to 1449
(3) Find the average of odd numbers from 9 to 579
(4) Find the average of odd numbers from 15 to 1003
(5) What is the average of the first 1574 even numbers?
(6) What is the average of the first 1307 even numbers?
(7) Find the average of even numbers from 12 to 610
(8) Find the average of odd numbers from 5 to 735
(9) What is the average of the first 1143 even numbers?
(10) What is the average of the first 283 even numbers?