Question:
Find the average of odd numbers from 13 to 629
Correct Answer
321
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 629
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 629 are
13, 15, 17, . . . . 629
After observing the above list of the odd numbers from 13 to 629 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 629 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 629
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 629
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 629
= 13 + 629/2
= 642/2 = 321
Thus, the average of the odd numbers from 13 to 629 = 321 Answer
Method (2) to find the average of the odd numbers from 13 to 629
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 629 are
13, 15, 17, . . . . 629
The odd numbers from 13 to 629 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 629
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 629
629 = 13 + (n – 1) × 2
⇒ 629 = 13 + 2 n – 2
⇒ 629 = 13 – 2 + 2 n
⇒ 629 = 11 + 2 n
After transposing 11 to LHS
⇒ 629 – 11 = 2 n
⇒ 618 = 2 n
After rearranging the above expression
⇒ 2 n = 618
After transposing 2 to RHS
⇒ n = 618/2
⇒ n = 309
Thus, the number of terms of odd numbers from 13 to 629 = 309
This means 629 is the 309th term.
Finding the sum of the given odd numbers from 13 to 629
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 629
= 309/2 (13 + 629)
= 309/2 × 642
= 309 × 642/2
= 198378/2 = 99189
Thus, the sum of all terms of the given odd numbers from 13 to 629 = 99189
And, the total number of terms = 309
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 629
= 99189/309 = 321
Thus, the average of the given odd numbers from 13 to 629 = 321 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 954
(2) Find the average of even numbers from 6 to 256
(3) Find the average of the first 1516 odd numbers.
(4) Find the average of the first 2210 odd numbers.
(5) What is the average of the first 1110 even numbers?
(6) Find the average of the first 3585 odd numbers.
(7) Find the average of even numbers from 10 to 844
(8) Find the average of even numbers from 8 to 118
(9) Find the average of odd numbers from 9 to 1081
(10) What will be the average of the first 4087 odd numbers?