Question : Find the average of odd numbers from 15 to 579
Correct Answer 297
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 579
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 579 are
15, 17, 19, . . . . 579
After observing the above list of the odd numbers from 15 to 579 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 579 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 579
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 579
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 579
= 15 + 579/2
= 594/2 = 297
Thus, the average of the odd numbers from 15 to 579 = 297 Answer
Method (2) to find the average of the odd numbers from 15 to 579
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 579 are
15, 17, 19, . . . . 579
The odd numbers from 15 to 579 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 579
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 579
579 = 15 + (n – 1) × 2
⇒ 579 = 15 + 2 n – 2
⇒ 579 = 15 – 2 + 2 n
⇒ 579 = 13 + 2 n
After transposing 13 to LHS
⇒ 579 – 13 = 2 n
⇒ 566 = 2 n
After rearranging the above expression
⇒ 2 n = 566
After transposing 2 to RHS
⇒ n = 566/2
⇒ n = 283
Thus, the number of terms of odd numbers from 15 to 579 = 283
This means 579 is the 283th term.
Finding the sum of the given odd numbers from 15 to 579
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 579
= 283/2 (15 + 579)
= 283/2 × 594
= 283 × 594/2
= 168102/2 = 84051
Thus, the sum of all terms of the given odd numbers from 15 to 579 = 84051
And, the total number of terms = 283
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 579
= 84051/283 = 297
Thus, the average of the given odd numbers from 15 to 579 = 297 Answer
Similar Questions
(1) Find the average of the first 3337 odd numbers.
(2) Find the average of the first 4393 even numbers.
(3) Find the average of the first 3737 even numbers.
(4) Find the average of even numbers from 4 to 1724
(5) Find the average of the first 2462 odd numbers.
(6) Find the average of the first 3131 even numbers.
(7) Find the average of the first 524 odd numbers.
(8) What is the average of the first 1841 even numbers?
(9) Find the average of the first 3235 odd numbers.
(10) Find the average of odd numbers from 11 to 629