Question:
Find the average of odd numbers from 3 to 1215
Correct Answer
609
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 1215
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 1215 are
3, 5, 7, . . . . 1215
After observing the above list of the odd numbers from 3 to 1215 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 1215 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 1215
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1215
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 3 to 1215
= 3 + 1215/2
= 1218/2 = 609
Thus, the average of the odd numbers from 3 to 1215 = 609 Answer
Method (2) to find the average of the odd numbers from 3 to 1215
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 1215 are
3, 5, 7, . . . . 1215
The odd numbers from 3 to 1215 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1215
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 3 to 1215
1215 = 3 + (n – 1) × 2
⇒ 1215 = 3 + 2 n – 2
⇒ 1215 = 3 – 2 + 2 n
⇒ 1215 = 1 + 2 n
After transposing 1 to LHS
⇒ 1215 – 1 = 2 n
⇒ 1214 = 2 n
After rearranging the above expression
⇒ 2 n = 1214
After transposing 2 to RHS
⇒ n = 1214/2
⇒ n = 607
Thus, the number of terms of odd numbers from 3 to 1215 = 607
This means 1215 is the 607th term.
Finding the sum of the given odd numbers from 3 to 1215
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 3 to 1215
= 607/2 (3 + 1215)
= 607/2 × 1218
= 607 × 1218/2
= 739326/2 = 369663
Thus, the sum of all terms of the given odd numbers from 3 to 1215 = 369663
And, the total number of terms = 607
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 3 to 1215
= 369663/607 = 609
Thus, the average of the given odd numbers from 3 to 1215 = 609 Answer
Similar Questions
(1) Find the average of the first 3904 odd numbers.
(2) Find the average of odd numbers from 13 to 173
(3) Find the average of the first 4679 even numbers.
(4) Find the average of the first 4157 even numbers.
(5) What will be the average of the first 4131 odd numbers?
(6) Find the average of odd numbers from 3 to 707
(7) Find the average of the first 3164 even numbers.
(8) Find the average of the first 2524 even numbers.
(9) Find the average of even numbers from 8 to 510
(10) Find the average of odd numbers from 15 to 815