Question:
Find the average of odd numbers from 7 to 1207
Correct Answer
607
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 1207
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 1207 are
7, 9, 11, . . . . 1207
After observing the above list of the odd numbers from 7 to 1207 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 1207 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 1207
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 1207
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 1207
= 7 + 1207/2
= 1214/2 = 607
Thus, the average of the odd numbers from 7 to 1207 = 607 Answer
Method (2) to find the average of the odd numbers from 7 to 1207
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 1207 are
7, 9, 11, . . . . 1207
The odd numbers from 7 to 1207 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 1207
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 1207
1207 = 7 + (n – 1) × 2
⇒ 1207 = 7 + 2 n – 2
⇒ 1207 = 7 – 2 + 2 n
⇒ 1207 = 5 + 2 n
After transposing 5 to LHS
⇒ 1207 – 5 = 2 n
⇒ 1202 = 2 n
After rearranging the above expression
⇒ 2 n = 1202
After transposing 2 to RHS
⇒ n = 1202/2
⇒ n = 601
Thus, the number of terms of odd numbers from 7 to 1207 = 601
This means 1207 is the 601th term.
Finding the sum of the given odd numbers from 7 to 1207
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 1207
= 601/2 (7 + 1207)
= 601/2 × 1214
= 601 × 1214/2
= 729614/2 = 364807
Thus, the sum of all terms of the given odd numbers from 7 to 1207 = 364807
And, the total number of terms = 601
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 1207
= 364807/601 = 607
Thus, the average of the given odd numbers from 7 to 1207 = 607 Answer
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