Question:
Find the average of odd numbers from 15 to 1521
Correct Answer
768
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1521
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1521 are
15, 17, 19, . . . . 1521
After observing the above list of the odd numbers from 15 to 1521 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1521 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1521
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1521
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 1521
= 15 + 1521/2
= 1536/2 = 768
Thus, the average of the odd numbers from 15 to 1521 = 768 Answer
Method (2) to find the average of the odd numbers from 15 to 1521
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1521 are
15, 17, 19, . . . . 1521
The odd numbers from 15 to 1521 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1521
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 1521
1521 = 15 + (n – 1) × 2
⇒ 1521 = 15 + 2 n – 2
⇒ 1521 = 15 – 2 + 2 n
⇒ 1521 = 13 + 2 n
After transposing 13 to LHS
⇒ 1521 – 13 = 2 n
⇒ 1508 = 2 n
After rearranging the above expression
⇒ 2 n = 1508
After transposing 2 to RHS
⇒ n = 1508/2
⇒ n = 754
Thus, the number of terms of odd numbers from 15 to 1521 = 754
This means 1521 is the 754th term.
Finding the sum of the given odd numbers from 15 to 1521
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 1521
= 754/2 (15 + 1521)
= 754/2 × 1536
= 754 × 1536/2
= 1158144/2 = 579072
Thus, the sum of all terms of the given odd numbers from 15 to 1521 = 579072
And, the total number of terms = 754
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 1521
= 579072/754 = 768
Thus, the average of the given odd numbers from 15 to 1521 = 768 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 287
(2) Find the average of the first 3138 even numbers.
(3) Find the average of the first 2455 odd numbers.
(4) What is the average of the first 434 even numbers?
(5) Find the average of the first 1371 odd numbers.
(6) Find the average of odd numbers from 15 to 103
(7) Find the average of odd numbers from 7 to 1213
(8) Find the average of odd numbers from 11 to 41
(9) Find the average of the first 3767 even numbers.
(10) Find the average of the first 4800 even numbers.