Question : Find the average of odd numbers from 7 to 419
Correct Answer 213
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 419
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 419 are
7, 9, 11, . . . . 419
After observing the above list of the odd numbers from 7 to 419 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 419 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 419
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 419
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 7 to 419
= 7 + 419/2
= 426/2 = 213
Thus, the average of the odd numbers from 7 to 419 = 213 Answer
Method (2) to find the average of the odd numbers from 7 to 419
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 419 are
7, 9, 11, . . . . 419
The odd numbers from 7 to 419 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 419
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 7 to 419
419 = 7 + (n – 1) × 2
⇒ 419 = 7 + 2 n – 2
⇒ 419 = 7 – 2 + 2 n
⇒ 419 = 5 + 2 n
After transposing 5 to LHS
⇒ 419 – 5 = 2 n
⇒ 414 = 2 n
After rearranging the above expression
⇒ 2 n = 414
After transposing 2 to RHS
⇒ n = 414/2
⇒ n = 207
Thus, the number of terms of odd numbers from 7 to 419 = 207
This means 419 is the 207th term.
Finding the sum of the given odd numbers from 7 to 419
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 7 to 419
= 207/2 (7 + 419)
= 207/2 × 426
= 207 × 426/2
= 88182/2 = 44091
Thus, the sum of all terms of the given odd numbers from 7 to 419 = 44091
And, the total number of terms = 207
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 7 to 419
= 44091/207 = 213
Thus, the average of the given odd numbers from 7 to 419 = 213 Answer
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