Question:
Find the average of odd numbers from 5 to 589
Correct Answer
297
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 589
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 589 are
5, 7, 9, . . . . 589
After observing the above list of the odd numbers from 5 to 589 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 589 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 589
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 589
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 589
= 5 + 589/2
= 594/2 = 297
Thus, the average of the odd numbers from 5 to 589 = 297 Answer
Method (2) to find the average of the odd numbers from 5 to 589
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 589 are
5, 7, 9, . . . . 589
The odd numbers from 5 to 589 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 589
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 589
589 = 5 + (n – 1) × 2
⇒ 589 = 5 + 2 n – 2
⇒ 589 = 5 – 2 + 2 n
⇒ 589 = 3 + 2 n
After transposing 3 to LHS
⇒ 589 – 3 = 2 n
⇒ 586 = 2 n
After rearranging the above expression
⇒ 2 n = 586
After transposing 2 to RHS
⇒ n = 586/2
⇒ n = 293
Thus, the number of terms of odd numbers from 5 to 589 = 293
This means 589 is the 293th term.
Finding the sum of the given odd numbers from 5 to 589
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 589
= 293/2 (5 + 589)
= 293/2 × 594
= 293 × 594/2
= 174042/2 = 87021
Thus, the sum of all terms of the given odd numbers from 5 to 589 = 87021
And, the total number of terms = 293
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 589
= 87021/293 = 297
Thus, the average of the given odd numbers from 5 to 589 = 297 Answer
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